 61 solved out of 314 shown
$500 If$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert=n$is such that the subset sums$\sum_{a\in A'}a$are distinct for all$A'\subseteq A$then $N \gg 2^{n}.$ Erdős called this 'perhaps my first serious problem'. The powers of$2$show that$2^n$would be best possible here. The trivial lower bound is$N \gg 2^{n}/n$, since all$2^n$distinct subset sums must lie in$\{1,\ldots,Nn\}$. Erdős and Moser [Er56] proved $N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.$ A number of improvements of the constant have been given (see [St23] for a history), with the current record$\sqrt{2/\pi}$due to Dubroff, Fox, and Xu [DFX21]. In [ErGr80] the generalisation where$A\subseteq (0,N]$is a set of real numbers such that the subset sums all differ by at least$1$is proposed, with the same conjectured bound. Additional thanks to: Zachary Hunter$1000
Can the smallest modulus of a covering system be arbitrarily large?
Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15] has shown the answer is no: the smallest modulus must be at most $10^{18}$.
$5000 If$A\subseteq \mathbb{N}$has$\sum_{n\in A}\frac{1}{n}=\infty$then must$A$contain arbitrarily long arithmetic progressions? This is essentially asking for good bounds on$r_k(N)$, the size of the largest subset of$\{1,\ldots,N\}$without a non-trivial$k$-term arithmetic progression. For example, a bound like $r_k(N) \ll_k \frac{N}{(\log N)(\log\log N)^2}$ would be sufficient. Even the case$k=3$is non-trivial, but was proved by Bloom and Sisask [BlSi20]. Much better bounds for$r_3(N)$were subsequently proved by Kelley and Meka [KeMe23]. Green and Tao [GrTa17] have proved$r_4(N)\ll N/(\log N)^{c}$for some small constant$c>0$. The best bound available for general$k$is due to Gowers [Go01], $r_k(N) \ll \frac{N}{(\log\log N)^{c_k}},$ where$c_k>0$is a small constant depending on$k$. Curiously, Erdős [Er83c] thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao [GrTa08]. See also .$10000
For any $C>0$ are there infinitely many $n$ such that $p_{n+1}-p_n> C\frac{\log\log n\log\log\log\log n}{(\log\log \log n)^2}\log n?$
The peculiar quantitative form of Erdős' question was motivated by an old result of Rankin [Ra38], which showed the existence of some constant $C>0$ such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FoGrKoTa16]. The best bound available, due to all five authors [FoGrKoMaTa18], is that there are infinitely many $n$ such that $p_{n+1}-p_n\gg \frac{\log\log n\log\log\log\log n}{\log\log \log n}\log n.$ The likely truth is a lower bound like $\gg(\log n)^2$. In [Er97c] Erdős revised the value of this problem to \$5000 and reserved the \$10000 for a lower bound of $>(\log n)^{1+c}$ for some $c>0$.
Let $C\geq 0$. Is there an infinite sequence of $n_i$ such that $\lim_{i\to \infty}\frac{p_{n_i+1}-p_{n_i}}{\log n_i}=C?$
Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$. This problem asks whether $S=[0,\infty]$. Although this conjecture remains unproven, a lot is known about $S$. Some highlights:
• $\infty\in S$ by Westzynthius' result [We31] on large prime gaps,
• $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps,
• Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure,
• Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers,
• Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$,
• Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$,
• Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$, and that $S$ has bounded gaps.
$100 Let$d_n=p_{n+1}-p_n$. Are there infinitely many$n$such that$d_n<d_{n+1}<d_{n+2}$? Conjectured by Erdős and Turán [ErTu48]. Shockingly Erdős offered \$25000 for a disproof of this, but as he comments, it 'is certainly true'.

Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that $d_n<d_{n+1}<\cdots <d_{n+m}$ and infinitely many $n$ such that $d_n> d_{n+1}>\cdots >d_{n+m}.$

Is there a covering system all of whose moduli are odd?
Asked by Erdős and Selfridge (sometimes also with Schinzel). They also ask can there be such that no two of the moduli divide each other, or where all the moduli are odd and square-free?

Selfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$.

Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST21] have proved that if the moduli are all squarefree then at least one must be even.

For any finite colouring of the integers is there a covering system all of whose moduli are monochromatic?
Conjectured by Erdős and Graham, who also ask about a density-type version: for example, is $\sum_{\substack{a\in A\\ a>N}}\frac{1}{a}\gg \log N$ a sufficient condition for $A$ to contain the moduli of a covering system? The answer (to both colouring and density versions) is no, due to the result of Hough [Ho15] on the minimum size of a modulus in a covering system - in particular one could colour all integers $<10^{18}$ different colours and all other integers a new colour.
Let $A$ be the set of all integers not of the form $p+2^{k}+2^l$ (where $k,l\geq 0$ and $p$ is prime). Is the upper density of $A$ positive?
Crocker [Cr71] has proved there are are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$. Erdős believes this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.
Is there some $k$ such that every integer is the sum of a prime and at most $k$ powers of 2?
Erdős described this as 'probably unattackable'. In [ErGr80] Erdős and Graham suggest that no such $k$ exists. Gallagher [Ga75] has shown that for any $\epsilon>0$ there exists $k(\epsilon)$ such that the set of integers which are the sum of a prime and at most $k(\epsilon)$ many powers of 2 has lower density at least $1-\epsilon$.
Is every odd number the sum of a squarefree number and a power of 2?
Odlyzko has checked this up to $10^7$. Granville and Soundararajan [GrSo98] have proved that this is very related to the problem of finding primes $p$ for which $2^p\equiv 2\pmod{p^2}$ (for example this conjecture implies there are infinitely many such $p$).

Erdős thinks that proving this with two powers of 2 is perhaps easy.

Let $A$ be an infinite set such that there are no distinct $a,b,c\in A$ such that $a\mid (b+c)$. Is there such an $A$ with $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}>0?$
Asked by Erdős and Sárközy, who proved that $A$ must have density $0$. The set of $p^2$, where $p\equiv 3\pmod{4}$ is prime, has this property. Note that such a set cannot contain a three-term arithmetic progression, and hence by the bound of Kelley and Meka [KeMe23] we have $\lvert A\cap\{1,\ldots,N\}\rvert\ll \exp(-O((\log N)^{1/12}))N$ for all large $N$.
$100 Let$A\subseteq \{1,\ldots,N\}$be such that there are no$a<b<c\in A$such that$a\mid(b+c)$. Is it true that$\lvert A\rvert\leq N/3+O(1)$? Asked by Erdős and Sárközy, who observed that$[2N/3,N]\cap \mathbb{N}$is such a set. The answer is yes, as proved by Bedert [Be23]. Let$A\subseteq \mathbb{N}$. Let$B\subseteq \mathbb{N}$be the set of integers which are representable in exactly one way as the sum of two elements from$A$. Is it true that for all$\epsilon>0$and large$N$$\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}.$ Asked by Erdős, Sárközy, and Szemerédi, who constructed a set which shows this would be the best possible. Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where$p_n$is the sequence of primes? Erdős suggested that a computer could be used to explore this, and did not see any other method to attack this. Tao [Ta23] has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture. Is the set of odd integers not of the form$2^k+p$the union of an infinite arithmetic progression and a set of density$0$? Erdős called this conjecture 'rather silly'. Using covering congruences Erdős [Er50] proved that the set of such odd integers contains an infinite arithmetic progression. Are there infinitely many primes$p$such that every even number$n\leq p-3$can be written as a difference of primes$n=q_1-q_2$where$q_1,q_2\leq p$? The first prime without this property is$97$. We call$m$practical if every integer$n<m$is the sum of distinct divisors of$m$. If$m$is practical then let$h(m)$be such that$h(m)$many divisors always suffice. Are there infinitely many practical$m$such that $h(m) < (\log\log m)^{O(1)}?$ Is it true that$h(n!)<n^{o(1)}$? It is easy to see that almost all numbers are not practical. This may be true with$m=n!$. Erdős originally showed that$h(n!) <n$. Vose [Vo85] has shown that$h(n!)\ll n^{1/2}$. Related to . Let$n_1<n_2<\cdots$be an arbitrary sequence of integers, each with an associated residue class$a_i\pmod{n_i}$. Let$A$be the set of integers$n$such that for every$i$either$n<n_i$or$n\not\equiv a_i\pmod{n_i}$. Must the logarithmic density of$A$exist? Let$A\subset\mathbb{N}$be infinite. Must there exist some$k\geq 1$such that almost all integers have a divisor of the form$a+k$for some$a\in A$? Asked by Erdős and Tenenbaum. Ruzsa has found a counterexample. Tenenbaum asked the weaker variant (still open) where for every$\epsilon>0$there is some$k=k(\epsilon)$such that at least$1-\epsilon$density of all integers have a divisor of the form$a+k$for some$a\in A$.$100
An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\ldots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$. Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$-almost covering system with $N\leq n_1$ and $n_k\leq Cn_1$?
By a simple averaging argument the set of moduli $[m_1,m_2]\cap \mathbb{N}$ has a choice of residue classes which form an $\epsilon(m_1,m_2)$-almost covering system with $\epsilon(m_1,m_2)=\prod_{m_1\leq m\leq m_2}(1-1/m).$ A $0$-covering system is just a covering system, and so by Hough [Ho15] these only exist for $n_1<10^{18}$. [NOTE: This is my best attempt at recovering problem 5 from [Er95], which doesn't make sense as written.]
$500 If$A\subseteq \mathbb{N}$is such that$A+A$contains all but finitely many integers then$\limsup 1_A\ast 1_A(n)=\infty$. Conjectured by Erdős and Turán. They also suggest the stronger conjecture that$\limsup 1_A\ast 1_A(n)/\log n>0$. Another stronger conjecture would be that$\lvert A\cap [1,N]\rvert \gg N^{1/2}$for all large$N$implies$\limsup 1_A\ast 1_A(n)=\infty$. Erdős and Sárközy conjectured the stronger version that if$A=\{a_1<a_2<\cdots\}$and$B=\{b_1<b_2<\cdots\}$with$a_n/b_n\to 1$such that$A+B=\mathbb{N}$then$\limsup 1_A\ast 1_B(n)=\infty$.$100
Is there an explicit construction of a set $A\subseteq \mathbb{N}$ such that $A+A=\mathbb{N}$ but $1_A\ast 1_A(n)=o(n^\epsilon)$ for every $\epsilon>0$?
The existence of such a set was asked by Sidon to Erdős in 1932. Erdős (eventually) proved the existence of such a set using probabilistic methods. This problem asks for a constructive solution.
$1000 Let$h(N)$be the maximum size of a Sidon set in$\{1,\ldots,N\}$. Is it true that, for every$\epsilon>0$, $h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?$ It may even be true that$h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of$N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to$0.998N^{1/4}$, which has been further optimised by O'Bryant [OB22] to yield $h(N)\leq N^{1/2}+0.99703N^{1/4}$ for sufficiently large$N$. Additional thanks to: Zachary Hunter Given any infinite set$A\subset \mathbb{N}$there is a set$B$of density$0$such that$A+B$contains all except finitely many integers. Proved by Lorentz [Lo54]. Is there a set$A\subset\mathbb{N}$such that $\lvert A\cap\{1,\ldots,N\}\rvert = o((\log N)^2)$ and such that every large integer can be written as$p+a$for some prime$p$and$a\in A$? Erdős [Er54] proved by the probabilistic method that such a set$A$exists with$\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$. Let$A\subset\mathbb{N}$be such that every large integer can be written as$n^2+a$for some$a\in A$and$n\geq 0$. What is the smallest possible value of $\limsup \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}?$ Erdős observed that this value is finite and$>1$. Additional thanks to: Timothy Gowers Is there$A\subset\mathbb{N}$such that every large integer can be written as$2^n+a$for some$a\in A$and$n\geq 0$and $\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N/\log N}<\infty?$ Let$A\subseteq\mathbb{N}$be an additive basis of order$k$. Is it true that for every$B\subseteq\mathbb{N}$we have $d_s(A+B)\geq \alpha+\frac{\alpha(1-\alpha)}{k},$ where$\alpha=d_s(A)$and $d_s(A) = \inf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N}$ is the Schnirelmann density? Erdős proved this is true with$k$replaced by$2k$in the denominator. Find the optimal constant$c>0$such that the following holds. For all sufficiently large$N$, if$A\sqcup B=\{1,\ldots,2N\}$is a partition into two equal parts, so that$\lvert A\rvert=\lvert B\rvert=N$, then$\| 1_A\circ 1_B\|_\infty \geq cN$- that is, there is some$x$such that the number of solutions to$a-b=x$with$a\in A$and$b\in B$is at least$cN$. The minimum overlap problem. The example (with$N$even)$A=\{N/2+1,\ldots,3N/2\}$shows that$c\leq 1/2$(indeed, Erdős initially conjectured that$c=1/2$). The lower bound of$c\geq 1/4$is trivial, and Scherk improved this to$1-1/\sqrt{2}=0.29\cdots$. The current records are $0.379005 < c < 0.380926\cdots,$ the lower bound due to White [Wh22] and the upper bound due to Haugland [Ha16]. We say that$A\subset \mathbb{N}$is an essential component if$d_s(A+B)>d_s(B)$for every$B\subset \mathbb{N}$with$0<d_s(B)<1$where$d_s$is the Schnirelmann density. Can a lacunary set$A\subset\mathbb{N}$(i.e. there exists some$\lambda>1$such that$A=\{n_1<n_2<\cdots\}$satisfies$n_{k+1}>\lambda n_k$) be an essential component? The answer is no by Ruzsa [Ru87], who proved that if$A$is an essential component then there exists some constant$c>0$such that$\lvert A\cap \{1,\ldots,N\}\rvert \geq (\log N)^{1+c}$for all large$N$. Is there$A\subset\mathbb{N}$which is not an additive basis, but is such that for every set$B\subseteq\mathbb{N}$of density$\beta$and every$N$there exists$a\in A$such that $\frac{\lvert B\cap (B+a)\cap \{1,\ldots,N\}\rvert}{N}\geq (\beta+f(\beta))N$ where$f(\beta)>0$for$0<\beta <1 $. Is there an infinite Sidon set$A\subset \mathbb{N}$such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all$\epsilon>0$? The trivial greedy construction achieves$\gg N^{1/3}$. The current best bound of$\gg N^{\sqrt{2}-1+o(1)}$is due to Ruzsa [Ru98]. Erdős proved that for every infinite Sidon set$A$we have $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0,$ and also that there is a set$A\subset \mathbb{N}$with$\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$such that$1_A\ast 1_A(n)=O(1)$.$500
For what functions $g(N)\to \infty$ is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}g(N)$ implies $\limsup 1_A\ast 1_A(n)=\infty$?
It is possible that this is true even with $g(N)=O(1)$, from which the Erdős-Turán conjecture would follow.
$500 Let$A\subset\mathbb{N}$be an infinite set such that the triple sums$a+b+c$are all distinct for$a,b,c\in A$(aside from the trivial coincidences). Is it true that $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=0?$ Is it true that $\limsup \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/3}}=\infty?$ Erdős proved that if the pairwise sums$a+b$are all distinct aside from the trivial coincidences then $\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.$ Let$M\geq 1$and$N$be sufficiently large in terms of$M$. Is it true that for every maximal Sidon set$A\subset \{1,\ldots,N\}$there is another Sidon set$B\subset \{1,\ldots,N\}$of size$M$such that$(A-A)\cap(B-B)=\{0\}$?$100
If $A,B\subset \{1,\ldots,N\}$ are two Sidon sets such that $(A-A)\cap(B-B)=\{0\}$ then is it true that $\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq\binom{f(N)}{2}+O(1),$ where $f(N)$ is the maximum possible size of a Sidon set in $\{1,\ldots,N\}$? If $\lvert A\rvert=\lvert B\rvert$ then can this bound be improved to $\binom{\lvert A\rvert}{2}+\binom{\lvert B\rvert}{2}\leq (1-c)\binom{f(N)}{2}$ for some constant $c>0$?
Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M=M(\epsilon)$ and $B\subset \{N+1,\ldots,M\}$ such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?
Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?
This follows from the colouring result of Croot [Cr03]. Obtaining better quantitative aspects seems interesting still -- note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
The answer is yes, as proved by Croot [Cr03].
$100 If$\delta>0$and$N$is sufficiently large in terms of$\delta$, and$A\subseteq\{1,\ldots,N\}$is such that$\sum_{a\in A}\frac{1}{a}>\delta \log N$then must there exist$S\subseteq A$such that$\sum_{n\in S}\frac{1}{n}=1$? Solved by Bloom [Bl21], who showed that the quantitative threshold $\sum_{n\in A}\frac{1}{n}\gg \frac{\log\log\log N}{\log\log N}\log N$ is sufficient. Erdős further speculates that perhaps even$\gg (\log\log N)^2$might be sufficient. (A construction of Pomerance, as discussed in the appendix of [Bl21], shows that this would be best possible.) Are there infinitely many integers$n,m$such that$\phi(n)=\sigma(m)$? This would follow immediately from the twin prime conjecture. The answer is yes, proved by Ford, Luca, and Pomerance [FLP10]. Let$A=\{a_1<\cdots<a_t\}\subseteq \{1,\ldots,N\}$be such that$\phi(a_1)<\cdots<\phi(a_t)$. The primes are such an example. Are they the largest possible? Can one show that$\lvert A\rvert<(1+o(1))\pi(N)$or even$\lvert A\rvert=o(N)$? Erdős remarks that the last conjecture is probably easy, and that similar questions can be asked about$\sigma(n)$. Solved by Tao [Ta23b], who proved that $\lvert A\rvert \leq \left(1+O\left(\frac{(\log\log x)^5}{\log x}\right)\right)\pi(x).$$250
Schoenberg proved that for every $c\in [0,1]$ the density of $\{ n\in \mathbb{N} : \phi(n)<cn\}$ exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive?
Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$?
Carmichael has asked whether there is an integer $t$ for which $\phi(n)=t$ has exactly one solution. Erdős has proved that if such a $t$ exists then there must be infinitely many such $t$.
$250 Let$A$be a finite set of integers. Is it true that for every$\epsilon>0$$\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?$ The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of$\lvert A\rvert^{1+c}$for some constant$c>0$, and an upper bound of$o(\lvert A\rvert^2)$. The lower bound has been improved a number of times. The current record is $\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}$ due to Rudnev and Stevens [RuSt22] (note$1558/1167=1.33504\cdots$). Let$A$be a finite set of integers. Is it true that, for every$k$, if$\lvert A\rvert$is sufficiently large depending on$k$, then there are least$\lvert A\rvert^k$many integers which are either the sum or product of distinct elements of$A$? Asked by Erdős and Szemerédi [ErSz83]. Solved by Chang [Ch03].$100
A set of integers $A$ is Ramsey $2$-complete if, whenever $A$ is $2$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. It is known that it cannot be true that $\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^2$ for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$ $\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.$ Improve either of these bounds.
The stated bounds are due to Burr and Erdős [BuEr85].
$250 A set of integers$A$is Ramsey$r$-complete if, whenever$A$is$r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of$A$. Prove any non-trivial bounds about the growth rate of such an$A$for$r>2$. A paper of Burr and Erdős [BuEr85] proves both upper and lower bounds for$r=2$, showing that there exists some$c>0$such that it cannot be true that $\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2$ for all large$N$, and also constructing a Ramsey$2$-complete$A$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^3.$ Burr has shown that the sequence of$k$th powers is Ramsey$r$-complete for every$r,k\geq 1$. Solved by Conlon, Fox, and Pham [CFP21], who constructed for every$r\geq 2$an$r$-Ramsey complete$A$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,$ and showed that this is best possible, in that there exists some constant$c>0$such that if$A\subset \mathbb{N}$satisfies $\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2$ for all large$N$then$A$cannot be$r$-Ramsey complete. Additional thanks to: Mehtaab Sawhney Suppose$A\subseteq \{1,\ldots,N\}$is such that there are no$k+1$elements of$A$which are relatively prime. An example is the set of all multiples of the first$k$primes. Is this the largest such set? This was disproved for$k\geq 8$by Khachatrian [AhKh94]. Erdős asks if the conjecture remains true provided$n\geq (1+o(1))p_k^2$.$500
Is there $A\subseteq \mathbb{N}$ such that $\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$ exists and is $\neq 0$?
A suitably constructed random set has this property if we are allowed to ignore an exceptional set of density zero. The challenge is obtaining this with no exceptional set. Erdős believed the answer should be no. Erdős and Sárkzözy proved that $\frac{\lvert 1_A\ast 1_A(n)-\log n\rvert}{\sqrt{\log n}}\to 0$ is impossible. Erdős suggests it may even be true that the $\liminf$ and $\limsup$ of $1_A\ast 1_A(n)/\log n$ are always separated by some absolute constant.
Is $\sum_{n\geq 2}\frac{1}{n!-1}$ irrational?
Is $\sum_{n\geq 2}\frac{\omega(n)}{2^n}$ irrational? (Here $\omega(n)$ counts the number of distinct prime divisors of $n$.)
Erdős [Er48] proved that $\sum_n \frac{d(n)}{2^n}$ is irrational, where $d(n)$ is the divisor function.
Is every integer $n\equiv 0\pmod{4}$ equal to $2^k+m$ for some $k\geq 0$ and squarefree integer $m$?
Erdős could prove this is true for almost all $n\equiv 0\pmod{4}$.
Let $F_{k}(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square. Is $F_5(N)=(1-o(1))N$? More generally, is $F_{2k+1}(N)=(1-o(1))N$?
Conjectured by Erdős, Sós, and Sárkzözy [ErSaSo95], who proved this for $F_3(N)$. Erdős [Er35] earlier proved that $F_4(N)=o(N)$. Erdős also asks about $F(N)$, the size of the largest such set such that the product of no odd number of $a\in A$ is a square. Ruzsa proved that $\lim F(N)/N <1$. The value of $\lim F(N)/N$ is unknown, but it is $>1/2$.
Let $f(n)$ be a number theoretic function which grows slowly and $F(n)$ such that for almost all $n<x$ we have $f(n)/F(n)\to 0$. When are there infinitely many $x$ such that $\frac{\#\{ n\in \mathbb{N} : n+f(n)\in (x,x+F(x))\}}{F(x)}\to \infty?$
Conjectured by Erdős, Pomerance, and Sárközy [ErPoSa97] prove this when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erdős believes it is false when $f(n)=\phi(n)$ or $\sigma(n)$.
$250 Let$a,b,c$be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form$a^kb^lc^m$($k,l,m\geq 0$), none of which divide any other? Conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when$a=3$,$b=5$, and$c=7$. Let$d_1<d_2<\cdots <d_k$be integers such that $\sum_{1\leq i\leq k}\frac{1}{d_i-1}\geq 1.$ Can all sufficiently large integers be written as a sum of the form$\sum_{1\leq i\leq k}a_i$, where$a_i$has only the digits$0,1$when written in base$d_i$? Conjectured by Burr, Erdős, Graham, and Li [BuErGrLi96]. Pomerance observed that the condition$\sum 1/(d_i-1)\geq 1$is necessary. In [BuErGrLi96] they prove the property holds for$\{3,4,7\}$. Additional thanks to: Boris Alexeev and Dustin Mixon Let$A\subseteq \mathbb{N}$be the set of integers which have only the digits$0,1$when written base$3$, and$B\subseteq \mathbb{N}$be the set of integers which have only the digits$0,1$when written base$4$. Does$A+B$have positive density? Conjectured by Burr, Erdős, Graham, and Li [BuErGrLi96].$250
Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?
Investigated by Erdős and Turán [ErTu34] in their first joint paper, where they proved that $\log n \ll f(n) \ll n/\log n$ (the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erdős says that $f(n)=o(n/\log n)$ has never been proved, but perhaps never seriously attacked.
Let $\epsilon>0$ and $N$ be sufficiently large depending on $\epsilon$. Is there $A\subseteq\{1,\ldots,N\}$ such that no $a\in A$ divides the sum of any distinct elements of $A\backslash\{a\}$ and $\lvert A\rvert\gg N^{1/2-\epsilon}$?
It is easy to see that we must have $\lvert A\rvert \ll N^{1/2}$. Csaba has constructed such an $A$ with $\lvert A\rvert \gg N^{1/5}$.
Let $k>3$. Does the product of any $k$ consecutive integers $N=\prod_{m< n\leq m+k}n$ (with $m>k$) have a prime factor $p\mid N$ such that $p^2\nmid N$?
Erdős and Selfridge proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.
$250 The density of integers which have two divisors$d_1,d_2$such that$d_1<d_2<2d_1$exists and is equal to$1$. The answer is yes (in fact with$2$replaced with any constant$c>1$), proved by Maier and Tenenbaum [MaTe84]. See also . Let$F(k)$be the number of solutions to $1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},$ where$1\leq n_1<\cdots<n_k$are distinct integers. Find good estimates for$F(k)$. The current best bounds known are $2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},$ where$c>0$is some absolute constant and$c_0=1.26408\cdots$is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21]. A set$A\subset \mathbb{N}$is primitive if no member of$A$divides another. Is the sum $\sum_{n\in A}\frac{1}{n\log n}$ maximised over all primitive sets when$A$is the set of primes? Erdős [Er35] proved that this sum always converges for a primitive set. Solved by Lichtman [Li23]. Show that, for any$n\geq 5$, the binomial coefficient$\binom{2n}{n}$is not squarefree. Proved by Sárkzözy [Sa85] for all sufficiently large$n$, and by Granville and Ramaré [GrRa96] for all$n\geq 5$. More generally, if$f(n)$is the largest integer such that, for some prime$p$, we have$p^{f(n)}$dividing$\binom{2n}{n}$, then$f(n)$should tend to infinity with$n$. Can one even disprove that$f(n)\gg \log n$? Is it true that all sufficiently large$n$can be written as$2^k+m$for some$k\geq 0$, where$\Omega(m)<\log\log m$? (Here$\Omega(m)$is the number of prime divisors of$m$counted with multiplicity.) What about$<\epsilon \log\log m$? Or some more slowly growing function? It is easy to see by probabilistic methods that this holds for almost all integers. Romanoff [Ro34] showed that a positive density set of integers are representable as the sum of$2^k+p$for prime$p$, and Erdős used covering systems to show that there is a positive density set of integers which cannot be so represented. Let$x\in \overline{\mathbb{Q}}$be an algebraic number. For any$n\geq 1$let $R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x$ be the maximal sum of$n$distinct unit fractions which is$<x$. Is it true that, for sufficiently large$n$, we have $R_{n+1}(x)=R_n(x)+\frac{1}{m},$ where$m$is minimal such that$m$does not appear in$R_n(x)$and the right-hand side is$<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?) If$x$is irrational then this can be false, but it may be true for almost all$x\in\mathbb{R}$. Curtiss [Cu22] showed that this is true for$x=1$and Erdős [Er50b] showed it is true for all$x\in\mathbb{Q}$. Let$s_1<s_2<\cdots$be the sequence of squarefree numbers. Is it true that, for any$\epsilon>0$and large$n$, $s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?$ Is it true that $s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?$ Erdős [Er51] showed that there are infinitely many$n$such that $s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n},$ so this bound would be the best possible. The current best bound is due to Chan [Ch21], who proved an upper bound of$s_n^{\frac{5}{26}+o(1)}$. More precisely, they proved that there exists some$C>0$such that, for all large$x$, the interval $(x,x+Cx^{5/26}]$ contains a squarefree number. See also . Let$d_n=p_{n+1}-p_n$. The set of$n$such that$d_{n+1}\geq d_n$has density$1/2$, and similarly for$d_{n+1}\leq d_n$. Furthermore, there are infinitely many$n$such that$d_{n+1}=d_n$. Are there arbitrarily long arithmetic progressions of primes? The answer is yes, proved by Green and Tao [GrTa08]. The stronger claim that there are arbitrarily long arithmetic progressions of consecutive primes is still open. Let$n\geq 1$and $A=\{a_1<\cdots <a_{\phi(n)}\}=\{ 1\leq m<n : (m,n)=1\}.$ Is it true that $\sum_{1\leq k<\phi(n)}(a_{k+1}-a_k)^2 \ll \frac{n^2}{\phi(n)}?$ The answer is yes, as proved by Montgomery and Vaughan [MoVa86], who in fact found the correct order of magnitude with the power$2$replaced by any$\gamma\geq 1$. Is there a set$A\subset\mathbb{N}$such that, for all large$N$, $\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N$ and such that every large integer can be written as$2^k+a$for some$k\geq 0$and$a\in A$? Lorentz [Lo54] showed there is such a set with, for all large$N$, $\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N$ Let$n_1<n_2<\cdots$be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences$n_{k+1}-n_k$. Erdős [Er51] proved that, for infinitely many$k$, $n_{k+1}-n_k \gg \frac{\log n_k}{\sqrt{\log\log n_k}}.$ Richards [Ri82] improved this to $\limsup_{k\to \infty} \frac{n_{k+1}-n_k}{\log n_k} \geq 1/4.$ The constant$1/4$here has been improved, most lately to$0.868\cdots$by Dietmann, Elsholtz, Kalmynin, Konyagin, and Maynard [DEKKM22]. The best known upper bound is due to Bambah and Chowla [BaCh47], who proved that $n_{k+1}-n_k \ll n_k^{1/4}.$ Let$d_n=p_{n+1}-p_n$. Prove that $\sum_{1\leq n\leq N}d_n^2 \ll N(\log N)^2.$ Cramer proved an upper bound of$O(N(\log N)^4)$conditional on the Riemann hypothesis. The prime number theorem immediately implies a lower bound of$\gg N(\log N)^2$. For every$c\geq 0$the density$f(c)$of integers for which $\frac{p_{n+1}-p_n}{\log n}< c$ exists and is a continuous function of$c$. Let$N_k=2\cdot 3\cdots p_k$and$\{a_1<a_2<\cdots <a_{\phi(N_k)}\}$be the integers$<N_k$which are relatively prime to$N_k$. Then, for any$c\geq 0$, the limit $\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}$ exists and is a continuous function of$c$. Let$f(n)$count the number of solutions to$n=p+2^k$for prime$p$and$k\geq 0$. Is it true that$f(n)=o(\log n)$? Erdős [Er50] proved that there are infinitely many$n$such that$f(n)\gg \log\log n$. Erdős could not even prove that there do not exist infinitely many integers$n$such that for all$2^k<n$the number$n-2^k$is prime (probably$n=105$is the largest such integer). Let$A\subseteq \mathbb{N}$be a set such that$\lvert A\cap \{1,\ldots,N\}\rvert \gg \log N$for all large$N$. Let$f(n)$count the number of solutions to$n=p+a$for$p$prime and$a\in A$. Is it true that$\limsup f(n)=\infty$? Erdős [Er50] proved this when$A=\{2^k : k\geq 0\}$. Solved by Chen and Ding [ChDi22]. Let$c_1,c_2>0$. Is it true that, for any sufficiently large$x$, there exist more than$c_1\log x$many consecutive primes$\leq x$such that the difference between any two is$>c_2$? This is well-known if$c_1$is sufficiently small. Let$f:\mathbb{N}\to \{-1,1\}$be a multiplicative function. Is it true that $\lim_{N\to \infty}\frac{1}{N}\sum_{n\leq N}f(n)$ always exists? Wintner observed that if$f$can take complex values on the unit circle then the limit need not exist. The answer is yes, as proved by Halász [Ha68]. Is there an infinite set of primes$P$such that if$\{a_1<a_2<\cdots\}$is the set of integers divisible only by primes in$P$then$\lim a_{i+1}-a_i=\infty$? Originally asked to Erdős by Wintner. The limit is infinite for a finite set of primes, which follows from a theorem of Pólya. Additional thanks to: Boris Alexeev and Dustin Mixon For every$n\geq 2$there exist distinct integers$1\leq x<y<z$such that $\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ The Erdős-Straus conjecture. The existence of a representation of$4/n$as the sum of at most four distinct unit fractions follows trivially from a greedy algorithm. Schinzel conjectured the generalisation that, for any fixed$a$, if$n$is sufficiently large in terms of$a$then there exist distinct integers$1\leq x<y<z$such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Let$a_1<a_2<\cdots$be a sequence of integers such that $\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1$ and$\sum\frac{1}{a_n}\in \mathbb{Q}$. Then, for all sufficiently large$n\geq 1$, $a_n = a_{n-1}^2-a_{n-1}+1.$ A sequence defined in such a fashion is known as Sylvester's sequence. Let$C>1$. Does the set of integers of the form$p+\lfloor C^k\rfloor$, for some prime$p$and$k\geq 0$, have density$>0$? Originally asked to Erdős by Kalmár. Erdős believed the answer is yes. Romanoff [Ro34] proved that the answer is yes if$C$is an integer. Let$(a,b)=1$. Every large integer is the sum of distinct integers of the form$a^kb^l$with$k,l\geq 0$. Proved by Birch [Bi59]. Let$n_1<n_2<\cdots$be a sequence of integers such that $\limsup \frac{n_k}{k}=\infty.$ Is $\sum_{k=1}^\infty \frac{1}{2^{n_k}}$ transcendental? Erdős [Er75c] proved the answer is yes under the stronger condition that$\limsup n_k/k^t=\infty$for all$t\geq 1$. Are there infinitely many$n$such that, for all$k\geq 1$, $\omega(n+k) \ll k,$ where the implied constant depends only on$n$? (Here$\omega(n)$is the number of distinct prime divisors of$n$.) Related to . Erdős and Graham [ErGr80] write 'we just know too little about sieves to be able to handle such a question ("we" here means not just us but the collective wisdom (?) of our poor struggling human race).' Is $\sum_n \frac{\phi(n)}{2^n}$ irrational? Here$\phi$is the Euler totient function. Is $\sum \frac{\sigma(n)}{2^n}$ irrational? (Here$\sigma(n)$is the sum of divisors function.) The answer is yes, as shown by Nesterenko [Ne96]. Is $\sum \frac{p_n}{2^n}$ irrational? (Here$p_n$is the$n$th prime.) Let$k\geq 1$and$\sigma_k(n)=\sum_{d\mid n}d^k$. Is $\sum \frac{\sigma_k(n)}{n!}$ irrational? This is known now for$1\leq k\leq 4$. The cases$k=1,2$are reasonably straightforward, as observed by Erdős [Er52]. The case$k=3$was proved independently by Schlage-Puchta [ScPu06] and Friedlander, Luca, and Stoiciu [FLC07]. The case$k=4$was proved by Pratt [Pr22]. Let$a_1<a_2<\cdots $be an infinite sequence of integers such that$a_{i+1}/a_i\to 1$. If every arithmetic progression contains infinitely many integers which are the sum of distinct$a_i$then every sufficiently large integer is the sum of distinct$a_i$. This was disproved by Cassels [Ca60]. Let$A\subseteq \mathbb{N}$be such that $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert \to \infty\textrm{ as }x\to \infty$ and $\sum_{n\in A} \{ \theta n\}=\infty$ for every$\theta\in (0,1)$, where$\{x\}$is the distance of$x$from the nearest integer. Then every sufficiently large integer is the sum of distinct elements of$A$. Cassels [Ca60] proved this under the alternative hypotheses $\lvert A\cap [1,2x]\rvert -\lvert A\cap [1,x]\rvert\gg \log\log x$ and $\sum_{n\in A} \{ \theta n\}^2=\infty$ for every$\theta\in (0,1)$. Are there infinitely many$n$such that there exists some$t\geq 2$and$a_1,\ldots,a_t\geq 1$such that $\frac{n}{2^n}=\sum_{1\leq k\leq t}\frac{a_k}{2^{a_k}}?$ Is this true for all$n$? Is there a rational$x$such that $x = \sum_{k=1}^\infty \frac{a_k}{2^{a_k}}$ has at least two solutions? Related to . Additional thanks to: Zachary Chase Let$X\subseteq \mathbb{R}^3$be the set of all points of the shape $\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1},\sum_{n\in A} \frac{1}{n+2}\right)$ as$A\subseteq\mathbb{N}$ranges over all infinite sets with$\sum_{n\in A}\frac{1}{n}<\infty$. Does$X$contain an open set? Erdős and Straus proved the answer is yes for the 2-dimensional version, where$X\subseteq \mathbb{R}^2$is the set of $\left( \sum_{n\in A} \frac{1}{n},\sum_{n\in A}\frac{1}{n+1}\right)$ as$A\subseteq\mathbb{N}$ranges over all infinite sets with$\sum_{n\in A}\frac{1}{n}<\infty$. Is there a covering system all of whose moduli are of the form$p-1$for some primes$p\geq 5$? Selfridge has found an example using divisors of$360$if$p=3$is allowed. If a finite system of$r$congruences$\{ a_i\pmod{n_i} : 1\leq i\leq r\}$covers$2^r$consecutive integers then it covers all integers. This is best possible as the system$2^{i-1}\pmod{2^i}$shows. This was proved indepedently by Selfridge and Crittenden and Vanden Eynden [CrVE70]. Is there an infinite Lucas sequence$a_0,a_1,\ldots,$where$(a_0,a_1)=1$and$a_{n+2}=a_{n+1}+a_n$for$n\geq 0$such that all$a_k$are composite, and yet no integer has a common factor with every term of the sequence? Whether such a composite Lucas sequence even exists was open for a while, but using covering systems Graham [Gr64] showed that $a_0 = 1786772701928802632268715130455793$ $a_1 = 1059683225053915111058165141686995$ generate such a sequence. This problem asks whether one can have a composite Lucas sequence without 'an underlying system of covering congruences responsible'. For which$n$is there a covering system$\{ a_d\pmod{d} : d\mid n\}$such that if$x\equiv a_{d_1}\pmod{d_1}$and$x\equiv a_{d_2}\pmod{d_2}$then$(d_1,d_2)=1$? The density of such$n$is zero. Let$A=\{n_1<\cdots<n_r\}$be a finite set of integers. What is the minimum value of the density of integers not hit by a suitable choice of congreunces$a_i\pmod{n_i}$? Is the minimum achieved when all the$a_i$are equal? Let$k\geq 3$. Is there a choice of congruence classes$a_p\pmod{p}$for every prime$p$such that all except finitely many integers can be written as$a_p+tp$for some prime$p$and integer$t\geq k$? Even the case$k=3$seems difficult. This may be true with the primes replaced by any set$A\subseteq \mathbb{N}$such that $\lvert A\cap [1,N]\rvert \gg N/\log N$ and $\sum_{\substack{n\in A\\ n\leq N}}\frac{1}{n} -\log\log N\to \infty$ as$N\to \infty$. For$k=1$or$k=2$any set$A$such that$\sum_{n\in A}\frac{1}{n}=\infty$has this property. Let$n_1<n_2<\cdots $be an infinite sequence of integers with associated$a_i\pmod{n_i}$, such that for some$\epsilon>0$we have$n_k>(1+\epsilon)k\log k$for all$k$. Then $\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).$ Erdős and Graham [ErGr80] suggest this 'may have to await improvements in current sieve methods' (of which there have been several since 1980). Let$n_1<n_2<\cdots$be an infinite sequence with associated congruence classes$a_i\pmod{n_i}$such that the set of integers not satisfying any of the congruences$a_i\pmod{n_i}$has density$0$. Is it true that for every$\epsilon>0$there exists some$k$such that the density of integers not satisfying any of the congruences$a_i\pmod{n_i}$for$1\leq i\leq k$is less than$\epsilon$? The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies$\sum \frac{1}{n_i}=\infty$. Let$A\subseteq \mathbb{N}$be an infinite set and consider the following greedy algorithm for a rational$x\in (0,1)$: choose the minimal$n\in A$such that$n\geq 1/x$and repeat with$x$replaced by$x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of$x$as the sum of distinct unit fractions with denominators from$A$. Does this process always terminate if$x$has odd denominator and$A$is the set of odd numbers? More generally, for which pairs$x$and$A$does this process terminate? In 1202 Fibonacci observed that this process terminates for any$x$when$A=\mathbb{N}$. The problem when$A$is the set of odd numbers is due to Stein. Graham [Gr64b] has shown that$\frac{m}{n}$is the sum of distinct unit fractions with denominators$\equiv a\pmod{d}$if and only if $\left(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\right)=1.$ Does the greedy algorithm always terminate in such cases? Graham [Gr64c] has also shown that$x$is the sum of distinct unit fractions with square denominators if and only if$x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always. Additional thanks to: Zachary Hunter Let$p:\mathbb{Z}\to \mathbb{Z}$be a polynomial whose leading coefficient is positive and such that there exists no$d\geq 2$with$d\mid p(n)$for all$n\geq 1$. Is it true that, for all sufficiently large$m$, there exist integers$1\leq n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ and $m=p(n_1)+\cdots+p(n_k)?$ Graham [Gr63] has proved this when$p(x)=x$. Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of$p(n_i)$with distinct$n_i$. Burr has proved this if$p(x)=x^k$with$k\geq 1$and if we allow$n_i=n_j$. Let$f(k)$be the maximal value of$n_1$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{k}{e-1}?$ The upper bound$f(k) \leq (1+o(1))\frac{k}{e-1}$is trivial since for any$u\geq 1$we have $\sum_{u\leq n\leq eu}\frac{1}{n}=1+o(1),$ and hence if$f(k)=u$then we must have$k\geq (e-1-o(1))u$. Essentially solved by Croot [Cr01], who showed that for any$N>1$there exists some$k\geq 1$and $N<n_1<\cdots <n_k \leq (e+o(1))N$ with$1=\sum \frac{1}{n_i}$. Additional thanks to: Zachary Hunter Let$f(k)$be the minimal value of$n_k$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{e}{e-1}k?$ It is trivial that$f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any$u\geq 1$$\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),$ and so if$eu\approx f(k)$then$k\leq \frac{e-1}{e}f(k)$. Proved by Martin [Ma00]. Additional thanks to: Zachary Hunter Let$k\geq 2$. Is it true that there exists an interval$I$of width$(e-1+o(1))k$and integers$n_1<\cdots<n_k\in I$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ The answer is yes, proved by Croot [Cr01]. Let$k\geq 2$. Is it true that, for any distinct integers$n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ we must have$\max(n_{i+1}-n_i)\geq 3$? The example$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$shows that$3$would be best possible here. The lower bound of$\geq 2$is equivalent to saying that$1$is not the sum of reciprocals of consecutive integers, proved by Erdős [Er32]. This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large$N$, there exists a prime$p\in [N,2N]$such that$\frac{p+1}{2}$is also prime. Is it true that there are only finitely many pairs of intervals$I_1,I_2$such that $\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?$ This is still open even if$\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any$k$intervals. Is it true that, for all sufficiently large$k$, there exist intervals$I_1,\ldots,I_k$with$\lvert I_i\rvert \geq 2$for$1\leq i\leq k$such that $1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?$ Let$a\geq 1$. Must there exist some$b>a$such that $\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},$ with$(r_i,s_i)=1$and$s_2<s_1$? If so, how does this$b(a)$grow with$a$? For example, $\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.$ Let$n\geq 1$and define$L_n$to be the lowest common multiple of$\{1,\ldots,n\}$and$a_n$by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.$ Is it true that$(a_n,L_n)=1$and$(a_n,L_n)>1$both occur for infinitely many$n$? Let$A$be the set of$n\in \mathbb{N}$such that there exist$1\leq m_1<\cdots <m_k=n$with$\sum\tfrac{1}{m_i}=1$. Explore$A$. In particular, • Does$A$have density$1$? • What are those$n\in A$not divisible by any$d\in A$with$1<d<n$? Straus observed that$A$is closed under multiplication. Furthermore, it is easy to see that$A$does not contain any prime power. Additional thanks to: Zachary Hunter Let$k\geq 1$and let$v(k)$be the minimal integer which does not appear as some$n_i$in a solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$1\leq n_1<\cdots <n_k$. Estimate the growth of$v(k)$. Results of Bleicher and Erdős [BlEr75] imply$v(k) \gg k!$. It may be that$v(k)$grows doubly exponentially in$\sqrt{k}$or even$k$. An elementary inductive argument shows that$n_k\leq ku_k$where$u_1=1$and$u_{i+1}=u_i(u_i+1)$, and hence $v(k) \leq kc_0^{2^k},$ where $c_0=\lim_n u_n^{1/2^n}=1.26408\cdots$ is the 'Vardi constant'. Additional thanks to: Zachary Hunter Let$N\geq 1$and let$t(N)$be the least integer$t$such that there is no solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$t=n_1<\cdots <n_k\leq N$. Estimate$t(N)$. Erdős and Graham [ErGr80] claim a bound of $t(N)\ll\frac{N}{\log N},$ but have no idea of the true value of$t(N)$. Let$N\geq 1$and let$k(N)$denote the smallest$k$such that there exist$N\leq n_1<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $\lim_{N\to \infty} k(N)-(e-1)N=\infty?$ Erdős and Straus [ErSt71b] have proved the existence of some constant$c>0$such that $-c < k(N)-(e-1)N \ll \frac{N}{\log N}.$ Let$N\geq 1$and let$k(N)$be maximal such that there are$k$disjoint$A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$with$\sum_{n\in A_i}\frac{1}{n}=1$for all$i$. Estimate$k(N)$. Is it true that$k(N)=o(\log N)$? More generally, how many disjoint$A_i$can be found in$\{1,\ldots,N\}$such that the sums$\sum_{n\in A_i}\frac{1}{n}$are all equal? Using sunflowers it can be shown that there are at least$N\exp(-O(\sqrt{\log N}))$such sets. Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] (coupled with the trivial greedy approach) implies that$k(N)=(1-o(1))\log N$. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Let$N\geq 1$. How many$A\subseteq \{1,\ldots,N\}$are there such that$\sum_{n\in A}\frac{1}{n}=1$? It is not even known whether this is$2^{cN}$for some$c<1$or$2^{(1+o(1))N}$. Does every set$A\subseteq \mathbb{N}$of positive density contain some finite$S\subset A$such that$\sum_{n\in S}\frac{1}{n}=1$? The answer is yes, proved by Bloom [Bl21]. Is there an infinite sequence$a_1<a_2<\cdots $such that$a_{i+1}-a_i=O(1)$and no finite sum of$\frac{1}{a_i}$is equal to$1$? There does not exist such a sequence, which follows from the positive solution to  by Bloom [Bl21]. Let$A(N)$denote the maximal cardinality of$A\subseteq \{1,\ldots,N\}$such that$\sum_{n\in S}\frac{1}{n}\neq 1$for all$S\subseteq A$. Estimate$A(N)$. Erdős and Graham believe the answer is$A(N)=(1+o(1))N$. Croot [Cr03] disproved this, showing the existence of some constant$c<1$such that$A(N)<cN$for all large$N$. It is trivial that$A(N)\geq (1-\frac{1}{e}+o(1))N$. Additional thanks to: Zachary Hunter What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that for any$a,b_1,\ldots,b_k\in A$with$k\geq 2$we have $\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?$ Is there a constant$c>1/2$such that$\lvert A\rvert >cN$is possible for all large$N$? The example$A=(N/2,N]\cap \mathbb{N}$shows that$\lvert A\rvert\geq N/2$. Additional thanks to: Zachary Hunter Is it true that, for any$\delta>1/2$, if$N$is sufficiently large and$A\subseteq \{1,\ldots,N\}$has$\lvert A\rvert \geq \delta N$then there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}.$ The colouring version of this is , which was solved by Brown and Rödl [BrRo91]. The possible alternative question, that if$A\subseteq \mathbb{N}$is a set of positive lower density then must there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c},$ has a negative answer, taking for example$A$to be the union of$[5^k,(1+\epsilon)5^k]$for large$k$and sufficiently small$\epsilon>0$. This was observed by Hunter and Sawhney. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$ with distinct$a,b,c$? The density version of this is . This colouring version is true, as proved by Brown and Rödl [BrRo91]. For integers$1\leq a<b$let$N(a,b)$denote the minimal$k$such that there exist integers$1<n_1<\cdots<n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that$N(b) \ll \log\log b$? Erdős [Er50c] proved that $\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.$ The upper bound was improved by Vose [Vo85] to $N(b) \ll \sqrt{\log b}.$ One can also investigate the average of$N(a,b)$for fixed$b$, and it is known that $\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.$ Related to . Additional thanks to: Zachary Hunter For integers$1\leq a<b$let$D(a,b)$be the minimal value of$n_k$such that there exist integers$1\leq n_1<\cdots <n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that $D(b) \ll b(\log b)^{1+o(1)}?$ Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2.$ If$b=p$is a prime then $D(p) \gg p\log p.$ Let$a/b\in \mathbb{Q}_{>0}$with$b$squarefree. Are there integers$1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ For$n_i$the product of three distinct primes, this is true when$b=1$, as proved by Butler, Erdős and Graham [BuErGr15] (this paper is perhaps Erdős' last paper, appearing 19 years after his death). Are there two finite sets of primes$P,Q$such that $1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?$ Asked by Barbeau [Ba76]. Can this be done if we drop the requirement that all$p\in P$are prime and just ask for them to be relatively coprime, and similarly for$Q$? Let$N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Is it true that the set of integers not representable as such has the shape$[m,\infty)$for some$m$? Let$N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Are there$o(\log N)$such integers? Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] implies the answer to the second question is no: in fact all integers$m\leq (1-o(1))\log N$can be written as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Let$\alpha >0$and$N\geq 1$. Is it true that, for any$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert \geq \alpha N$there exists some$S\subseteq A$such that $\frac{a}{b}=\sum_{n\in S}\frac{1}{n}$ with$b =O_\alpha(1)$? What is the minimal value of$\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$as$A$ranges over all subsets of$\{1,\ldots,N\}$which contain no$S$such that$\sum_{n\in S}\frac{1}{n}=1$? Is it $e^{-(c+o(1))N}$ for some constant$c\in (0,1)$? It is trivially at least$1/[1,\ldots,N]$. Does there exist some$c>0$such that, for any$K>1$, whenever$A$is a sufficiently large finite multiset of integers with$\sum_{n\in A}\frac{1}{n}>K$there exists some$S\subseteq A$such that $1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?$ Erdős and Graham knew this with$e^{-cK}$replaced by$c/K^2$. Additional thanks to: Mehtaab Sawhney Are there infinitely many solutions to $\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},$ where$m\geq 2$is an integer and$p_1<\cdots<p_k$are distinct primes? For example, $\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}.$ Let$n\geq 1$and let$t$be minimal such that$\sum_{n\leq k\leq t}\frac{1}{k}\geq 1$. We define $\epsilon(n) = \sum_{n\leq k\leq t}\frac{1}{k}-1.$ How small can$\epsilon(n)$be? Is it true that $\liminf n^2\epsilon(n)=0?$ It is perhaps true that$n^{2+\delta}\epsilon(n)\to \infty$for all$\delta >0$. Let$u_1=1$and$u_{n+1}=u_n(u_n+1)$. Let$a_1<a_2<\cdots $be any other sequence with$\sum \frac{1}{a_k}=1$. Is it true that $\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\cdots?$$c_0$is called the Vardi constant. Is it true that if$A\subset \mathbb{N}\backslash\{1\}$is a finite set with$\sum_{n\in A}\frac{1}{n}<2$then there is a partition$A=A_1\sqcup A_2$such that $\sum_{n\in A_i}\frac{1}{n}<1$ for$i=1,2$? This is not true if$A$is a multiset, for example$2,3,3,5,5,5,5$. This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of$120$form a counterexample. More generally, Sándor shows that for any$n\geq 2$there exists a finite set$A\subseteq \mathbb{N}\backslash\{1\}$with$\sum_{k\in A}\frac{1}{k}<n$and no partition into$n$parts each of which has$\sum_{k\in A_i}\frac{1}{k}<1$. The minimal counterexample is$\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart. Additional thanks to: Tom Stobart Is there some constant$c>0$such that for every$n\geq 1$there exists some$\delta_k\in \{-1,0,1\}$for$1\leq k\leq n$with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$ Is it true that for sufficiently large$n$, for any$\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero? Inequality is obvious for the second claim, the problem is strict inequality. This fails for small$n$, for example $\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$ Additional thanks to: Zachary Chase Let$A\subseteq \mathbb{N}$be an infinite arithmetic progression and$f:A\to \{-1,1\}$be a non-constant function. Must there exist a finite$S\subset A$such that $\sum_{n\in S}\frac{f(n)}{n}=0?$ What about if$A$is an arbitrary set of positive density? What if$A$is the set of squares excluding$1$? Erdős and Straus [ErSt75] proved this when$A=\mathbb{N}$. Sattler [Sat75] proved this when$A$is the set of odd numbers. For the squares$1$must be excluded or the result is trivially false, since $\sum_{k\geq 2}\frac{1}{k^2}<2.$ What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that there is a function$\delta:A\to \{-1,1\}$such that $\sum_{n\in A}\frac{\delta_n}{n}=0$ and $\sum_{n\in A'}\frac{\delta_n}{n}\neq 0$ for all$A'\subsetneq A$? Let$S(N)$count the number of distinct sums of the form$\sum_{n\in A}\frac{1}{n}$for$A\subseteq \{1,\ldots,N\}$. Estimate$S(N)$. Bleicher and Erdős [BlEr75] proved the lower bound $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq \frac{\log S(N)}{\log 2},$ valid for$k\geq 4$and$\log_kN\geq k$, and also [BlEr76b] proved the upper bound $\log S(N)\leq \log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for$r\geq 1$and$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also . Additional thanks to: Boris Alexeev and Dustin Mixon What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that all sums$\sum_{n\in S}\frac{1}{n}$are distinct for$S\subseteq A$? Let$R(N)$be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for any$k\geq 4$with$\log_kN\geq k$and any$r\geq 1$with$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also . Additional thanks to: Boris Alexeev, Zachary Hunter, and Dustin Mixon Let$k\geq 3$and$A\subset \mathbb{N}$be the set of$k$th powers. What is the order of growth of$1_A^{(k)}(n)$, i.e. the number of representations of$n$as the sum of$k$many$k$th powers? Does there exist some$c>0$and infinitely many$n$such that $1_A^{(k)}(n) >n^c?$ Connected to Waring's problem. The famous Hypothesis$K$of Hardy and Littlewood was that$1_A^{(k)}(n)\leq n^{o(1)}$, but this was disproved by Mahler [Ma36] for$k=3$, who constructed infinitely many$n$such that $1_A^{(3)}(n)\gg n^{1/12}$ (where$A$is the set of cubes). Erdős believed Hypothesis$K$fails for all$k\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis$K^*$that for all$N$and$\epsilon>0$$\sum_{n\leq N}1_A^{(k)}(n)^2 \ll_\epsilon N^{1+\epsilon}.$ Erdős and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.' Independently Erdős [Er36] and Chowla proved that for all$k\geq 3$and infinitely many$n$$1_A^{(k)}(n) \gg n^{c/\log\log n}$ for some constant$c>0$(depending on$k$). Let$1\leq m\leq k$and$f_{k,m}(x)$denote the number of integers$\leq x$which are the sum of$m$many nonnegative$k$th powers. Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all$\epsilon>0$? Is it true that if$m<k$then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large$x$? This would have significant applications to Waring's problem. Erdős and Graham describe this as 'unattackable by the methods at our disposal'. The case$k=2$was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant$c>0$. For$k>2$it is not known if$f_{k,k}(x)=o(x)$. Does there exist a polynomial$f(x)$such that all the sums$f(a)+f(b)$with$a<b$nonnegative integers are distinct? Erdős and Graham describe this problem as 'very annoying'. Probably$f(x)=x^5$should work. Let$k\geq 3$and$f_{k,3}(x)$denote the number of integers$\leq x$which are the sum of three nonnegative$k$th powers. Is it true that $f_{k,3}(x) \gg x^{3/k}$ or even$\gg_\epsilon x^{3/k-\epsilon}$? Mahler and Erdős [ErMa38] proved that$f_{k,2}(x) \gg x^{2/k}$. For$k=3$the best known is due to Wooley [Wo15], $f_{3,3}(x) \gg x^{0.917\cdots}.$ Let$A\subset \mathbb{N}$be an additive basis of order$2$. Must there exist$B=\{b_1<b_2<\cdots\}\subseteq A$which is also a basis such that $\lim_{k\to \infty}\frac{b_k}{k^2}$ does not exist? Erdős originally asked whether this was true with$A=B$, but this was disproved by Cassels [Ca57]. Suppose$A\subseteq \{1,\ldots,N\}$is such that if$a,b\in A$then$a+b\nmid ab$. Can$A$be 'substantially more' than the odd numbers? What if$a,b\in A$with$a\neq b$implies$a+b\nmid 2ab$? Must$\lvert A\rvert=o(N)$? The connection to unit fractions comes from the observation that$\frac{1}{a}+\frac{1}{b}$is a unit fraction if and only if$a+b\mid ab$. Suppose$A\subseteq\mathbb{N}$and$C>0$is such that$1_A\ast 1_A(n)\leq C$for all$n\in\mathbb{N}$. Can$A$be partitioned into$t$many subsets$A_1,\ldots,A_t$(where$t=t(C)$depends only on$C$) such that$1_{A_i}\ast 1_{A_i}(n)<C$for all$1\leq i\leq t$and$n\in \mathbb{N}$? Asked by Erdős and Newman. Nešetřil and Rödl have shown the answer is no for all$C$(source is cited as 'personal communication' in [ErGr80]). Erdős had previously shown the answer is no for$C=3,4$and infinitely many other values of$C$. Suppose$A\subseteq \mathbb{N}$is a Sidon set. How large can $\limsup_{N\to \infty}\frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}$ be? Erdős proved that$1/2$is possible and Krückeberg [Kr61] proved$1/\sqrt{2}$is possible. Erdős and Turán [ErTu41] have proved this$\limsup$is always$\leq 1$. The fact that$1$is possible would follow if any finite Sidon set is a subset of a perfect difference set. Suppose$A\subset\mathbb{N}$is a minimal basis with positive density. Is it true that, for any$n\in A$, the (upper) density of integers which cannot be represented without using$n$is positive? Asked by Erdős and Nathanson. Let$A,B\subseteq \mathbb{N}$such that for all large$N$$\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}$ and $\lvert B\cap \{1,\ldots,N\}\rvert \gg N^{1/2}.$ Is it true that there are infinitely many solutions to$a_1-a_2=b_1-b_2\neq 0$with$a_1,a_2\in A$and$b_1,b_2\in B$? Let$A\subseteq \mathbb{N}$and$D(A)$be the set of those numbers which occur infinitely often as$a_1-a_2$with$a_1,a_2\in A$. What conditions on$A$are sufficient to ensure$D(A)$has bounded gaps? Prikry, Tijdeman, Stewart, and others (see the survey articles [St78] and [Ti79]) have shown that a sufficient condition is that$A$has positive density. One can also ask what conditions are sufficient for$D(A)$to have positive density, or for$\sum_{d\in D(A)}\frac{1}{d}=\infty$, or even just$D(A)\neq\emptyset$. Let$A\subseteq \mathbb{N}$be a set of density zero. Does there exist a basis$B$such that$A\subseteq B+B$and $\lvert B\cap \{1,\ldots,N\}\rvert =o(N^{1/2})$ for all large$N$? Erdős and Newman [ErNe77] have proved this is true when$A$is the set of squares. Find the best function$f(n)$such that every$n$can be written as$n=a+b$where both$a,b$are$f(n)$-smooth (that is, are not divisible by any prime$p>f(n)$.) Erdős originally asked if even$f(n)\leq n^{1/3}$is true. This is known, and the best bound is due to Balog [Ba89] who proved that $f(n) \ll_\epsilon n^{\frac{4}{9\sqrt{e}}+\epsilon}$ for all$\epsilon>0$. (Note$\frac{4}{9\sqrt{e}}=0.2695\ldots$.) It is likely that$f(n)\leq n^{o(1)}$, or even$f(n)\leq e^{O(\sqrt{\log n})}$. Additional thanks to: Zachary Hunter Let$d(A)$denote the density of$A\subseteq \mathbb{N}$. Characterise those$A,B\subseteq \mathbb{N}$with positive density such that $d(A+B)=d(A)+d(B).$ One way this can happen is if there exists$\theta>0$such that $A=\{ n>0 : \{ n\theta\} \in X_A\}\textrm{ and }B=\{ n>0 : \{n\theta\} \in X_B\}$ where$\{x\}$denotes the fractional part of$x$and$X_A,X_B\subseteq \mathbb{R}/\mathbb{Z}$are such that$\mu(X_A+X_B)=\mu(X_A)+\mu(X_B)$. Are all possible$A$and$B$generated in a similar way (using other groups)? For$r\geq 2$let$h(r)$be the maximal finite$k$such that there exists a basis$A\subseteq \mathbb{N}$of order$r$(so every large integer is the sum of at most$r$integers from$A$) and exact order$k$(i.e.$k$is minimal such that every large integer is the sum of exactly$k$integers from$A$). Find the value of $\lim_r \frac{h(r)}{r^2}.$ Erdős and Graham [ErGr80b] have shown that a basis$A$has an exact order if and only if$a_2-a_1,a_3-a_2,a_4-a_3,\ldots$are coprime. They also prove that $\frac{1}{4}\leq \lim_r \frac{h(r)}{r^2}\leq \frac{5}{4}.$ It is known that$h(2)=4$, but even$h(3)$is unknown (it is$\geq 7$). Additional thanks to: Zachary Hunter Let$A\subseteq \mathbb{N}$be a basis such that$\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $\lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert}{\lvert A\cap \{1,\ldots,N\}\rvert}=\infty?$ The restricted order of a basis is the least integer$t$(if it exists) such that every large integer is the sum of at most$t$distinct summands from$A$. What are necessary and sufficient conditions that this exists? Can it be bounded (when it exists) in terms of the order of the basis? What are necessary and sufficient conditions that this is equal to the order of the basis? Bateman has observed that for$h\geq 3$there is a basis of order$h$with no restricted order, taking $A=\{1\}\cup \{x>0 : h\mid x\}.$ Kelly [Ke57] has shown that any basis of order$2$has restricted order at most$4$and conjectures it always has restricted order at most$3$. The set of squares has order$4$and restricted order$5$(see [Pa33]) and the set of triangular numbers has order$3$and restricted order$3$(see [Sc54]). Is it true that if$A\backslash F$is a basis for all finite sets$F$then$A$must have a restricted order? What if they are all bases of the same order? Let$A\subseteq \mathbb{N}$be a basis of order$r$. Must the set of integers representable as the sum of exactly$r$distinct elements from$A$have positive density? Let$A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$be the greedy Sidon sequence: we begin with$1$and iteratively include the next smallest integer that preserves the Sidon property. What is the order of growth of$A$? Is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}$ for all$\epsilon>0$and large$N$? Erdős and Graham [ErGr80] also ask about the difference set$A-A$, whether this has positive density, and whether this contains$22$. This sequence is at OEIS A005282. Let$A=\{a_1<\cdots<a_k\}$be a finite set of integers and extend it to an infinite sequence$\overline{A}=\{a_1<a_2<\cdots \}$by defining$a_{n+1}$for$n\geq k$to be the least integer exceeding$a_n$which is not of the form$a_i+a_j$with$i,j\leq n$. Is it true that the sequence of differences$a_{m+1}-a_m$is eventually periodic? An old problem of Dickson. Even a starting set as small as$\{1,4,9,16,25\}$requires thousands of terms before periodicity occurs. With$a_1=1$and$a_2=2$let$a_{n+1}$for$n\geq 2$be the least integer$>a_n$which can be expressed uniquely as$a_i+a_j$for$i<j\leq n$. What can be said about this sequence? Do infinitely many pairs$a,a+2$occur? Does this sequence eventually have periodic differences? Is the density$0$? A problem of Ulam. The sequence is $1,2,3,4,6,8,11,13,16,18,26,28,\ldots$ at OEIS A002858. If$A\subseteq \mathbb{N}$is a multiset of integers such that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N$ for all$N$then must$A$be subcomplete? That is, must $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ contain an infinite arithmetic progression? A problem of Folkman. Folkman [Fo66] showed that this is true if $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1+\epsilon}$ for some$\epsilon>0$and all$N$. The original question was answered by Szemerédi and Vu [SzVu06] (who proved that the answer is yes). This is best possible, since Folkman [Fo66] showed that for all$\epsilon>0$there exists a multiset$A$with $\lvert A\cap \{1,\ldots,N\}\rvert\ll N^{1+\epsilon}$ for all$N$, such that$A$is not subcomplete. Additional thanks to: Zachary Hunter If$A\subseteq \mathbb{N}$is a set of integers such that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2}$ for all$N$then must$A$be subcomplete? That is, must $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ contain an infinite arithmetic progression? Folkman proved this under the stronger assumption that $\lvert A\cap \{1,\ldots,N\}\rvert\gg N^{1/2+\epsilon}$ for some$\epsilon>0$. This is true, and was proved by Szemerédi and Vu [SzVu06]. The stronger conjecture that this is true under $\lvert A\cap \{1,\ldots,N\}\rvert\geq (2N)^{1/2}$ seems to be still open (this would be best possible as shown by [Er61b]. Let$A\subseteq \mathbb{N}$be a complete sequence, and define the threshold of completeness$T(A)$to be the least integer$m$such that all$n\geq m$are in $P(A) = \left\{\sum_{n\in B}n : B\subseteq A\textrm{ finite }\right\}$ (the existence of$T(A)$is guaranteed by completeness). Is it true that there are infinitely many$k$such that$T(n^k)>T(n^{k+1})$? Erdős and Graham [ErGr80] remark that very little is known about$T(A)$in general. It is known that $T(n)=1, T(n^2)=128, T(n^3)=12758,$ $T(n^4)=5134240,\textrm{ and }T(n^5)=67898771.$ Erdős and Graham remark that a good candidate for the$n$in the question are$k=2^t$for large$t$, perhaps even$t=3$, because of the highly restricted values of$n^{2^t}$modulo$2^{t+1}$. Let$A=\{a_1< a_2<\cdots\}$be a set of integers such that •$A\backslash B$is complete for any finite subset$B$and •$A\backslash B$is not complete for any infinite subset$B$. Is it true that if$a_{n+1}/a_n \geq 1+\epsilon$for some$\epsilon>0$and all$n$then $\lim_n \frac{a_{n+1}}{a_n}=\frac{1+\sqrt{5}}{2}?$ Graham [Gr64d] has shown that the sequence$a_n=F_n-(-1)^{n}$, where$F_n$is the$n$th Fibonacci number, has these properties. Erdős and Graham [ErGr80] remark that it is easy to see that if$a_{n+1}/a_n>\frac{1+\sqrt{5}}{2}$then the second property is automatically satisfied, and that it is not hard to construct very irregular sequences satisfying both properties. Is there a sequence$A=\{a_1\leq a_2\leq \cdots\}$of integers with $\lim \frac{a_{n+1}}{a_n}=2$ such that $P(A')= \left\{\sum_{n\in B}n : B\subseteq A'\textrm{ finite }\right\}$ has density$1$for every cofinite subsequence$A'$of$A$? For what values of$0\leq m<n$is there a complete sequence$A=\{a_1\leq a_2\leq \cdots\}$of integers such that •$A$remains complete after removing any$m$elements, but •$A$is not complete after removing any$n$elements? The Fibonacci sequence$1,1,2,3,5,\ldots$shows that$m=1$and$n=2$is possible. The sequence of powers of$2$shows that$m=0$and$n=1$is possible. The case$m=2$and$n=3$is not known. For what values of$t,\alpha \in (0,\infty)$is the sequence$\lfloor t\alpha^n\rfloor$complete? Even in the range$t\in (0,1)$and$\alpha\in (1,2)$the behaviour is surprisingly complex. For example, Graham [Gr64e] has shown that for any$k$there exists some$t_k\in (0,1)$such that the set of$\alpha$such that the sequence is complete consists of at least$k$disjoint line segments. It seems likely that the sequence is complete for all$t>0$and all$1<\alpha < \frac{1+\sqrt{5}}{2}$. Proving this seems very difficult, since we do not even known whether$\lfloor (3/2)^n\rfloor$is odd or even infinitely often. If$A\subset\mathbb{N}$is a finite set of integers all of whose subset sums are distinct then $\sum_{n\in A}\frac{1}{n}<2.$ This was proved by Ryavec. The stronger statement that, for all$s\geq 0$, $\sum_{n\in A}\frac{1}{n^s} <\frac{1}{1-2^{-s}},$ was proved by Hanson, Steele, and Stenger [HSS77]. Let$p(x)\in \mathbb{Q}[x]$. Is it true that $A=\{ p(n)+1/n : n\in \mathbb{N}\}$ is strongly complete, in the sense that, for any finite set$B$, $\left\{\sum_{n\in X}n : X\subseteq A\backslash B\textrm{ finite }\right\}$ contains all sufficiently large rational numbers? Graham [Gr64f] proved this is true when$p(n)=n$. Erdős and Graham also ask which rational functions$r(x)\in\mathbb{Z}[x]$force$\{ r(n) : n\in\mathbb{N}\}$to be complete? Let$\alpha,\beta\in \mathbb{R}_{>0}$such that$\alpha/\beta$is irrational. Is $\{ \lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor,\lfloor 4\alpha\rfloor,\ldots\}\cup \{ \lfloor \beta\rfloor,\lfloor 2\beta\rfloor,\lfloor 4\beta\rfloor,\ldots\}$ complete? What if$2$is replaced by some$\gamma\in(1,2)$? Let$A\subseteq \mathbb{N}$be a lacunary sequence (so that$A=\{a_1<a_2<\cdots\}$and there exists some$\lambda>1$such that$a_{n+1}/a_n\geq \lambda$for all$n\geq 1$). Must $\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}$ contain all rationals in some open interval? Bleicher and Erdős conjecture the answer is no. Is there some$c>0$such that, for all sufficiently large$n$, there exist integers$a_1<\cdots<a_k\leq n$such that there are at least$cn^2$distinct integers of the form$\sum_{u\leq i\leq v}a_i$? This fails for$a_i=i$for example. Erdős and Graham also ask what happens if we drop the monotonicity restriction and just ask that the$a_i$are distinct. Perhaps some permutation of$\{1,\ldots,n\}$has at least$cn^2$such distinct sums. This latter problem has been solved by Konieczny [Ko15], who showed that a random permutation of$\{1,\ldots,n\}$has, with high probability, at least $\left(\frac{1+e^{-2}}{4}+o(1)\right) n^2$ many such consecutive sums. The original problem was solved (in the affirmative) by Beker [Be23b]. They also ask how many consecutive integers$>n$can be represented as such a sum? Is it true that, for any$c>0$at least$cn$such integers are possible (for sufficiently large$n$)? Additional thanks to: Adrian Beker Let$1\leq a_1<\cdots <a_k\leq n$are integers such that all sums of the shape$\sum_{u\leq i\leq v}a_i$are distinct. How large can$k$be? Must$k=o(n)$? Asked by Erdős and Harzheim. What if we remove the monotonicity and/or the distinctness constraint? Also what is the least$m$which is not a sum of the given form? Can it be much larger than$n$? Erdős and Harzheim can show that$\sum_{x<a_i<x^2}\frac{1}{a_i}\ll 1$. Is it true that$\sum_i \frac{1}{a_i}\ll 1$? Is there a sequence$A=\{1\leq a_1<a_2<\cdots\}$such that every integer is the sum of some finite number of consecutive elements of$A$? Can the number of representations of$n$in this form tend to infinity with$n$? Erdős and Moser [Mo63] considered the case when$A$is the set of primes, and conjectured that the$\limsup$of the number of such representations in this case is infinite. They could not even prove that the upper density of the set of integers representable in this form is positive. Let$a_1<a_2<\cdots$be an infinite sequence of integers such that$a_1=k$and$a_{i+1}$is the least integer which is not a sum of consecutive earlier$a_j$s. What can be said about the density of this sequence? A problem of MacMahon, studied by Andrews [An75]. Let$f(n)$be minimal such that$\{1,\ldots,n\}$can be partitioned into$f(n)$classes so that$n$cannot be expressed as a sum of distinct elements from the same class. How fast does$f(n)$grow? Erdős and Graham state it 'can be shown' that$f(n)\to \infty$. Let$c>0$and$n$be some large integer. What is the size of the largest$A\subseteq \{1,\ldots,\lfloor cn\rfloor\}$such that$n$is not a sum of a subset of$A$? Does this depend on$n$in an irregular way? Let$A\subseteq \mathbb{N}$be finite. Is it true that, for any fixed$t$, there are $\ll \frac{2^n}{n^{3/2}}$ many$S\subseteq A$such that$\sum_{n\in S}n=t$? Erdős and Moser proved this bound with an additional factor of$(\log n)^{3/2}$. This was removed by Sárközy and Szemerédi [SaSz65]. Stanley [St80] has shown that this quantity is maximised when$A$is an arithmetic progression and$t=\tfrac{1}{2}\sum_{n\in A}n$. When is the product of two or more disjoint blocks of consecutive integers a power? Is it true that there are only finitely many collections of disjoint intervals$I_1,\ldots,I_n$of size$\lvert I_i\rvert \geq 4$for$1\leq i\leq n$such that $\prod_{1\leq i\leq n}\prod_{n\in I_i}n$ is a square? Erdős and Selfridge have proved that the product of consecutive integers is never a power. The condition$\lvert I_i\rvert \geq 4$is necessary here, since Pomerance has observed that the product of $(2^{n-1}-1)2^{n-1}(2^{n-1}+1),$ $(2^n-1)2^n(2^n+1),$ $(2^{2n-1}-2)(2^{2n-1}-1)2^{2n-1},$ and $(2^{2n-2}-2)(2^{2n}-1)2^{2n}$ is a always a square. Are there any triples of consecutive integers all of which are powerful (i.e. if$p\mid n$then$p^2\mid n$)? Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. Do all pairs of consecutive powerful numbers come from solutions to Pell equations? Is the number of such pairs$\leq x$bounded by$(\log x)^{O(1)}$? Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. Are there any 2-full$n$such that$n+1$is 3-full? That is, if$p\mid n$then$p^2\mid n$and if$p\mid n+1$then$p^3\mid n+1$. Erdős originally asked Mahler whether there are infinitely many pairs of consecutive powerful numbers, but Mahler immediately observed that the answer is yes from the infinitely many solutions to the Pell equation$x^2=8y^2+1$. Note that$8$is 3-full and$9$is 2-full. Is the the only pair of such consecutive integers? Let$B_2(n)$be the 2-full part of$n$(that is,$B_2(n)=n/n'$where$n'$is the product of all primes that divide$n$exactly once). Is it true that, for every fixed$k\geq 1$, $\prod_{n\leq m<n+k}B_2(m) \ll n^{2+o(1)}?$ Or perhaps even$\ll_k n^2$? It would also be interesting to find upper and lower bounds for the analogous product with$B_r$for$r\geq 3$, where$B_r(n)$is the$r$-full part of$n$(that is, the product of prime powers$p^a \mid n$such that$p^{a+1}\nmid n$and$a\geq r$). Is it true that, for every fixed$r,k\geq 2$and$\epsilon>0$, $\limsup \frac{\prod_{n\leq m<n+k}B_r(m) }{n^{1+\epsilon}}\to\infty?$ How large is the largest prime factor of$n(n+1)$? Mahler [Ma35] showed that this is$\gg \log\log n$. Schinzel [Sc67b] observed that for infinitely many$n$it is$\leq n^{O(1/\log\log\log n)}$. The truth is probably$\gg (\log n)^2$for all$n$. Let$\epsilon>0$and$k\geq 2$. Is it true that, for all sufficiently large$n$, there is a sequence of$k$consecutive integers in$\{1,\ldots,n\}$all of which are$n^\epsilon$-smooth? Open even for$k=2$. Erdős and Graham remark 'the answer should be affirmative but the problem seems very hard'. Are there infinitely many$n$such that the largest prime factor of$n$is$<n^{1/2}$and the largest prime factor of$n+1$is$<(n+1)^{1/2}$? Pomerance has observed that if we replace$1/2$in the exponent by$1/\sqrt{e}-\epsilon$for any$\epsilon>0$then this is true for density reasons. Let$P(n)$denote the largest prime factor of$n$. Show that the set of$n$with$P(n)>P(n+1)$has density$1/2$. Conjectured by Erdős and Pomerance [ErPo78], who proved that this set and its complement both have positive upper density. Let$P(n)$denote the largest prime factor of$n$. There are infinitely many$n$such that$P(n)>P(n+1)>P(n+2)$. Conjectured by Erdős and Pomerance [ErPo78], who proved the analogous result for$P(n)<P(n+1)<P(n+2)$. Proved by Balog [Ba01], who proved that this is true for$\gg \sqrt{x}$many$n\leq x$(for all large$x$). Show that the equation $n! = a_1!a_2!\cdots a_k!,$ with$n-1>a_1\geq a_2\geq \cdots \geq a_k$, has only finitely many solutions. This would follow if$P(n(n+1))/\log n\to \infty$, where$P(m)$denotes the largest prime factor of$m$(see Problem ). Hickerson conjectured the largest solution is $16! = 14! 5!2!.$ The condition$a_1<n-1$is necessary to rule out the trivial solutions when$n=a_2!\cdots a_k!$. For any$m\in \mathbb{N}$, let$F(m)$be the minimal$k\geq 2$(if it exists) such that there are$a_1<\cdots <a_k=m$with$a_1!\cdots a_k!$a square. Let$D_k=\{ m : F(m)=k\}$. What is the order of growth of$\lvert D_k\cap\{1,\ldots,n\}\rvert$for$3\leq k\leq 6$? For example, is it true that$\lvert D_6\cap \{1,\ldots,n\}\rvert \gg n$? Studied by Erdős and Graham [ErGr76] (see also [LSS14]). It is known, for example, that: • no$D_k$contains a prime, •$D_2=\{ n^2 : n>1\}$, •$\lvert D_3\cap \{1,\ldots,n\}\rvert = o(\lvert D_4\cap \{1,\ldots,n\}\rvert)$, • the least element of$D_6$is$527$, and •$D_k=\emptyset$for$k>6$. Additional thanks to: Zachary Chase Let$n+1,\ldots,n+k$be consecutive composite integers. Are there distinct primes$p_1,\ldots,p_k$such that$p_i\mid n+i$for$1\leq i\leq k$? Very deep if true, since it would imply there is a prime between any two consecutive squares. Originally conjectured by Grimm [Gr69], who proved it when$k \ll \log n/\log\log n$. Ramachandra, Shorey, and Tijdeman [RST75] have improved this to $k \ll \left(\frac{\log n}{\log\log n}\right)^3.$ Are there infinitely many$n$such that$\binom{2n}{n}$is coprime to$105$? Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two primes$p$and$q$, there are infinitely many$n$such that$\binom{2n}{n}$is coprime to$pq$. Is $\sum_{\substack{p\nmid \binom{2n}{n}\\ p\leq n}}\frac{1}{p}$ unbounded as a function of$n$? If the sum is denoted by$f(n)$then Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n) = \sum_{k=2}^\infty \frac{\log k}{2^k}=\gamma_0$ and $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n)^2 = \gamma_0^2,$ so that for almost all integers$f(m)=\gamma_0+o(1)$. Let$r\geq 0$. Does the density of integers$n$for which$\binom{n}{k}$is squarefree for at least$r$values of$1\leq k<n$exist? Is this density$>0$? Erdős and Graham state they can prove that, for$k$fixed and large, the density of$n$such that$\binom{n}{k}$is squarefree is$o_k(1)$. They can also prove that there are infinitely many$n$such that$\binom{n}{k}$is not squarefree for$1\leq k<n$, and expect that the density of such$n$is positive. Let$S(n)$denote the largest integer such that, for all$1\leq k<n$, the binomial coefficient$\binom{n}{k}$is divisible by$p^{S(n)}$for some prime$p$(depending on$k$). Is it true that $\limsup S(n)=\infty?$ If$s(n)$denotes the largest integer such that$\binom{n}{k}$is divisible by$p^{s(n)}$for some prime$p$for at least one$1\leq k<n$then it is easy to see that$s(n)\to \infty$as$n\to \infty$(and in fact that$s(n) \asymp \log n$). We call an interval$[u,v]$'bad' if the greatest prime factor of$\prod_{u\leq m\leq v}m$occurs with an exponent greater than$1$. Let$B(x)$count the number of$n\leq x$which are contained in at least one bad interval. Is it true that $B(x)\sim \#\{ n\leq x: p\mid n\rightarrow p\leq n^{1/2}\}?$ Erdős and Graham only knew that$B(x) > x^{1-o(1)}$. Similarly, we call an interval$[u,v]$'very bad' if$\prod_{u\leq m\leq v}m$is powerful. The number of integers$n\leq x$contained in at least one very bad interval should be$\ll x^{1/2}$. In fact, it should be asymptotic to the number of powerful numbers$\leq x$. See also . The product of more than two consecutive integers is never powerful (a number$n$is powerful if whenever$p\mid n$we have$p^2\mid n$). Conjectured by Erdős and Selfridge. There are infinitely many$n$such that$n(n+1)$is powerful (see ). Let$u\leq v$be such that the largest prime dividing$\prod_{u\leq m\leq v}m$appears with exponent at least$2$. Is it true that$v-u=v^{o(1)}$? Can$v-u$be arbitrarily large? Erdős and Graham report it follows from results of Ramachandra that$v-u\leq v^{1/2+o(1)}$. See also . Is it true that for every$k$there are infinitely many primes$p$such that the largest prime divisor of $\prod_{0\leq i\leq k}(p^2+i)$ is$p$? If$1<k\leq n$then$\binom{n}{k}$is divisible by a prime$p<n/2$(except$\binom{7}{3}=5\cdot 7$). A conjecture of Erdős and Selfridge. Proved by Ecklund [Ec69], who made the stronger conjecture that whenever$n>k^2$the binomial coefficient$\binom{n}{k}$is divisible by a prime$p<n/k$. They have proved the weaker inequality$p\ll n/k^c$for some constant$c>0$. Let $F(n) = \max_{\substack{m<n\\ m\textrm{ composite}}} m+p(m),$ where$p(m)$is the least prime divisor of$m$. Is it true that$F(n)>n$for all sufficiently large$n$? Does$F(n)-n\to \infty$as$n\to\infty$? Can$\binom{n}{k}$be the product of consecutive primes infinitely often? For example $\binom{21}{2}=2\cdot 3\cdot 5\cdot 7.$ Erdős and Graham write that 'a proof that this cannot happen infinitely often for$\binom{n}{2}$seems hopeless; probably this can never happen for$\binom{n}{k}$if$3\leq k\leq n-3$.' Is there an absolute constant$c>0$such that, for all$1\leq k\leq n$, the binomial coefficient$\binom{n}{k}$always has a divisor in$(cn,n]$? Erdős once conjectured that$\binom{n}{k}$must always have a divisor in$(n-k,n]$, but this was disproved by Schinzel and Erdős [Sc58]. Can one classify all solutions of $\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)$ where$1<k_1<k_2$and$m_1+k_1\leq m_2$? Are there only finitely many solutions? More generally, if$k_1>2$then for fixed$a$and$b$$a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j)$ should have only a finite number of solutions. What if one just requires that the products have the same prime factors, say when$k_1=k_2$? Is it true that for every$n$there is a$k$such that $\prod_{1\leq i\leq k}(n+i) \mid \prod_{1\leq i\leq k}(n+k+i)?$ Asked by Erdős and Straus. For example when$n=2$we have$k=4$: $3\cdot 4\cdot 5\cdot 6 \mid 7\cdot 8\cdot 9\cdot 10.$ No example was known for$n\geq 10$. Let$f(n)$be the minimal$m$such that $n! = a_1\cdots a_k$ with$n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant$c$such that $f(n)-2n \sim c\frac{n}{\log n}?$ Erdős, Guy, and Selfridge [EGS82] have shown that $f(n)-2n \asymp \frac{n}{\log n}.$ Let$t(n)$be the maximum$m$such that $n!=a_1\cdots a_n$ with$m=a_1\leq \cdots \leq a_n$. Obtain good upper bounds for$t(n)$. In particular does there exist some constant$c>0$such that $t(n) \leq \frac{n}{e}-c\frac{n}{\log n}$ for infinitely many$n$? Erdős, Selfridge, and Straus have shown that $\lim \frac{t(n)}{n}=\frac{1}{e}.$ Alladi and Grinstead [AlGr77] have obtained similar results when the$a_i$are restricted to prime powers. Let$A(n)$denote the least value of$t$such that $n!=a_1\cdots a_t$ with$a_1\leq \cdots \leq a_t\leq n^2$. Is it true that $A(n)=\frac{n}{2}-\frac{n}{2\log n}+o\left(\frac{n}{\log n}\right)?$ If we change the condition to$a_t\leq n$it can be shown that $A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)$ via a greedy decomposition (use$n$as often as possible, then$n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the$a_t$. Let$f(n)$denote the minimal$m$such that $n! = a_1\cdots a_t$ with$a_1<\cdots <a_t=a_1+m$. What is the behaviour of$f(n)$? Erdős and Graham write that they do not even know whether$f(n)=1$infinitely often (i.e. whether a factorial is the product of two consecutive integers infinitely often). Let$t_k(n)$denote the least$m$such that $n\mid (m+1)(m+2).$ Is it true that $\sum_{1\leq n\leq N}t_2(n)=o(N)?$ The answer is yes, proved by Hall. It is probably true that the sum is$o(N/(\log N)^c)$for some constant$c>0$. Prove that there are no$n,x,y$with$x+n\leq y$such that $\mathrm{lcm}(x+1,\ldots,x+n) = \mathrm{lcm}(y+1,\ldots,y+n).$ It follows from the Thue-Siegel theorem that, for$n$fixed, there are only finitely many potential such$x$and$y$. Is it true that for every$k$there exists$n$such that $\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$ Erdős and Graham write that$n+1$always divides$\binom{2n}{n}$, but it is quite rare that$n$divides$\binom{2n}{n}$. Are there only finitely many solutions to $\prod_i \binom{2m_i}{m_i}=\prod_j \binom{2n_j}{n_j}$ with the$m_i,n_j$distinct? Are the only solutions to $n!=x^2-1$ when$n=4,5,7$? The Brocard-Ramanujan conjecture. Erdős and Graham describe this as an old conjecture, and write it 'is almost certainly true but it is intractable at present'. Overholt [Ov93] has shown that this has only finitely many solutions assuming a weak form of the abc conjecture. Is it true that there are no solutions to $n! = x^k\pm y^k$ when$k>2$? Erdős and Obláth [ErOb37] proved this is true when$(x,y)=1$and$k\neq 4$. Pollack and Shapiro [PoSh73] proved this is true when$(x,y)=1$and$k=4$. The known methods break down without the condition$(x,y)=1$. For any$k\geq 2$let$g_k(n)$denote the maximal value of $n-(a_1+\cdots+a_k)$ where$a_1,\ldots,a_k$are integers such that$a_1!\cdots a_k! \mid n!$. Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant$c_k$? Is it true that there is a constant$c_k$such that for almost all$n<x$we have $g_k(n)=c_k\log x+o(\log x)?$ Erdős and Graham write that it is easy to show that$g_k(n) \ll_k \log n$always, but the best possible constant is unknown. See also . Is there some function$\omega(r)$such that$\omega(r)\to \infty$as$r\to\infty$, such that for all large$n$there exist$a_1,a_2$with $a_1+a_2> n+\omega(r)\log n$ such that$a_1!a_2! \mid n!2^n3^n\cdots p_r^n$? See also . Prove that, for any finite set$A\subset\mathbb{N}$, there exist$a,b\in A$such that $\mathrm{gcd}(a,b)\leq a/\lvert A\rvert.$ A conjecture of Graham [Gr70], who also conjectured that (assuming$A$itself has no common divisor) then the only cases where equality achieved are when$A=\{1,\ldots,n\}$or$\{L/1,\ldots,L/n\}$(where$L=\mathrm{lcm}(1,\ldots,n)$) or$\{2,3,4,6\}$. Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87]. Proved for all sets by Balasubramanian and Soundararajan [BaSo96]. Does the equation $2^m=a_1!+\cdots+a_k!$ with$a_1<a_2<\cdots <a_k$have only finitely many solutions? Asked by Burr and Erdős. Frankl and Lin [Li76] independently showed that the answer is yes, and the largest solution is $2^7=2!+3!+5!.$ In fact Lin showed that the largest power of$2$which can divide a sum of distinct factorials containing$2$is$2^{254}$, and that there are only 5 solutions to$3^m=a_1!+\cdots+a_k!$(when$m=0,1,2,3,6$). See also . Let$f(a,p)$be the largest$k$such that there are$a=a_1<\cdots<a_k$such that $p^k \mid (a_1!+\cdots+a_k!).$ Is$f(a,p)$bounded by some absolute constant? What if this constant is allowed to depend on$a$and$p?

Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\sum_{i\leq k}a_i!$ then $m_k\to \infty$?

See also . Lin [Li76] has shown that $f(2,2) \leq 254$.
Let $p$ be a prime. Is it true that the equation $(p-1)!+a^{p-1}=p^k$ has only finitely many solutions?
Erdős and Graham remark that it is probably true that in general $(p-1)!+a^{p-1}$ is rarely a power at all (although this can happen, for example $6!+2^6=28^2$).
Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?
The only examples seem to be $4=1+3$ and $256=1+3+3^2+3^5$. If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.
Let $w(n)$ count the number of solutions to $n=2^a+3^b+2^c3^d$ with $a,b,c,d\geq 0$ integers. Is it true that $w(n)$ is bounded by some absolute constant?
A conjecture originally due to Newman.
Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ be the iterated $\phi$ function, so that $\phi_1(n)=\phi(n)$ and $\phi_k(n)=\phi(\phi_{k-1}(n))$. Let $f(n) = \min \{ k : \phi_k(n)=1\}.$ Does $f(n)/\log n$ have a distribution function? Is $f(n)/\log n$ almost always constant? What can be said about the largest prime factor of $\phi_k(n)$ when, say, $k=\log\log n$?
Pillai [Pi29] was the first to investigate this function, and proved $\log_3 n < f(n) < \log_2 n$ for all large $n$. Shapiro [Sh50] proved that $f(n)$ is essentially multiplicative.

Erdős, Granville, Pomerance, and Spiro [EGPS90] have proved that the answer to the first two questions is yes, conditional on a form of the Elliott-Halberstam conjecture.

It is likely true that, if $k\to \infty$ however slowly with $n$, then for almost $n$ the largest prime factor of $n$ is $\leq n^{o(1)}$.

How many iterations of $n\mapsto \phi(n)+1$ are needed before a prime is reached? Can infinitely many $n$ reach the same prime? What is the density of $n$ which reach any fixed prime?
A problem of Finucane. One can also ask about $n\mapsto \sigma(n)-1$.
Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that $\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?$
Let $g_1=g(n)=n+\phi(n)$ and $g_k(n)=g(g_{k-1}(n))$. For which $n$ and $r$ is it true that $g_{k+r}(n)=2g_k(n)$ for all large $k$?
The known solutions to $g_{k+2}(n)=2g_k(n)$ are $n=10$ and $n=94$. Selfridge and Weintraub found solutions to $g_{k+9}(n)=9g_k(n)$ and Weintraub found $g_{k+25}(3114)=729g_k(3114)$ for all $k\geq 6$.
Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that, for every $m,n$, there exist some $i,j$ such that $\sigma_i(m)=\sigma_j(n)$?
That is, there is (eventually) only one possible sequence that the iterated sum of divisors function can settle on. Selfridge reports numerical evidence suggests the answer is no, but Erdős and Graham write 'it seems unlikely that anything can be proved about this in the near future'.

Let $\nu(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\nu(m) \leq n$?
Erdős suggested this problem as a way of showing that the iterated behaviour of $n\mapsto n+\nu(n)$ eventually settles into a single sequence, regardless of the starting value of $n$ (see also  and ).

Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'. Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \nu(m)\leq n$ for all $m<n$?

Let $h_1(n)=h(n)=n+\tau(n)$ (where $\tau(n)$ counts the number of divisors of $n$) and $h_k(n)=h(h_{k-1}(n))$. Is it true, for any $m,n$, there exist $i$ and $j$ such that $h_i(m)=h_j(n)$?
Asked by Spiro. That is, there is (eventually) only one possible sequence that the iterations of $n\mapsto h(n)$ can settle on. Erdős and Graham believed the answer is yes. Similar questions can be asked by the iterates of many other functions. See also  and .
For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that $F(n)=(c+o(1))\log\log\log n$ for some constant $c$? Is the first pattern which fails to appear always $\phi(m+1)>\phi(m+2)>\cdots \phi(m+k)?$ Is it true that 'natural' ordering which mimics what happens to $\phi(1),\ldots,\phi(k)$ is the most likely to appear?
Erdős [Er36b] proved that $F(n)\asymp \log\log\log n,$ and similarly if we replace $\phi$ with $\sigma$ or $\tau$ or $\nu$ or any 'decent' additive or multiplicative function.
Let $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $V(2x)/V(x)\to 2$? Is there an asymptotic formula for $V(x)$?
Pillai [Pi29] proved $V(x)=o(x)$. The behaviour of $V(x)$ is almost completely understood. Maier and Pomerance [MaPo88] proved $V(x)=\frac{x}{\log x}e^{(C+o(1))\log\log\log x},$ for some explicit constant $C>0$. Ford [Fo98] improved this to $V(x)\asymp\frac{x}{\log x}e^{C_1(\log\log\log x-\log\log\log\log x)^2+C_2\log\log\log x-C_3\log\log\log\log x}$ for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\to 2$.

Let $V'(x)=\#\{\phi(m) : m\leq x\}$, and $V(x)$ count the number of $n\leq x$ such that $\phi(m)=n$ is solvable. Does $\lim V(x)/V'(x)$ exist? Is it $>1$?
It is trivial that $V'(x) \leq V(x)$. See also .
Are there infinitely many integers not of the form $n-\phi(n)$?
Asked by Erdős and Sierpiński. It follows from the Goldbach conjecture that every odd number can be written as $n-\phi(n)$. What happens for even numbers?

Erdős [Er73b] has shown that a positive density set of integers cannot be written as $\sigma(n)-n$.

If $\tau(n)$ counts the number of divisors of $n$, then what is the set of limit points of $\frac{\tau((n+1)!)}{\tau(n!)}?$
It can be shown that any number of the shape $1+1/k$ for $k\geq 1$ is a limit point, but no others are known.
If $\tau(n)$ counts the number of divisors of $n$ then let $F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!}{\tau(n!)}.$ Is it true that $\lim_{n\to \infty}F((\log n)^C,n)=\infty$ for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$? More generally, if $f(n)\leq \log n$ is a monotonic function then is $F(f,n)$ everywhere dense?
Erdős and Graham write that it is easy to show that $\lim F(n^{1/2},n)=\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.
Is there a sequence $1\leq d_1<d_2<\cdots$ with density $1$ such that all products $\prod_{u\leq i\leq v}d_i$ are distinct?
Let $f(1)=f(2)=1$ and for $n>2$ $f(n) = f(n-f(n-1))+f(n-f(n-2)).$ Does $f(n)$ miss infinitely many integers? What is its behaviour?
Asked by Hofstadter. The sequence begins $1,1,2,3,3,4,\ldots$ and is A005185 in the OEIS. It is not even known whether $f(n)$ is well-defined for all $n$.
Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ we choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?
Asked by Hofstadter. The sequence begins $1,2,3,5,6,8,10,11,\ldots$ and is A005243 in the OEIS.
Let $a_1=2$ and $a_2=3$ and continue the sequence by appending to $a_1,\ldots,a_n$ all possible values of $a_ia_j-1$ with $i\neq j$. Is it true that almost all integers appear in this sequence?
Asked by Hofstadter. The sequence begins $2,3,5,9,14,17,26,\ldots$ and is A005244 in the OEIS.
Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that $F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?$
Erdős [Er68] proved that $F(n)-\pi(n)\asymp n^{3/4}(\log n)^{-3/2}.$ Surprisingly, if we consider the corresponding problem in the reals (so the largest $m$ such that there are reals $1\leq a_1<\cdots <a_m\leq x$ such that for any distinct indices $i,j,r,s$ we have $\lvert a_ia_j-a_ra_s\rvert \geq 1$) then Alexander proved that $m> x/8e$ is possible.

Is there a set $A\subseteq \{1,\ldots,N\}$ of size $\ll N/\log N$ such that every $m\leq N$ can be written as $a+2^k$ for some $k\geq 0$ and $a\in A$?
The answer is yes, proved by Ruzsa [Ru72].
Is it true that, for every $n$ and $d$, there exists $k$ such that $d \mid p_{n+1}+\cdots+p_{n+k},$ where $p_r$ denotes the $r$th prime?
Is there a set $A\subseteq \mathbb{N}$ such that, for infinitely many $n$, all of $n-a$ are prime for all $a\in A$ with $0<a<n$ and $\liminf\frac{\lvert A\cap [1,x]\rvert}{\pi(x)}>0?$
Erdős and Graham could show this is true (assuming the prime $k$-tuple conjecture) if we replace $\liminf$ by $\limsup$.
Is it true that, if $A\subseteq \mathbb{N}$ is sparse enough and does not cover all residue classes modulo $p$ for any prime $p$, then there exists some $n$ such that $n+a$ is prime for all $a\in A$?
Fix some integer $n$ and define a decreasing sequence in $[1,n)$ by $a_1=n-1$ and, for $k\geq 2$, letting $a_k$ be the greatest integer in $[1,a_{k-1})$ such that all of the prime factors of $a_k$ are $>n-a_k$. Is it true that, for sufficiently large $n$, not all of this sequence can be prime?
Erdős and Graham write 'preliminary calculations made by Selfridge indicate that this is the case but no proof is in sight'. For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop.
Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the set of prime numbers up to finitely many exceptions?
A problem of Ostmann, sometimes known as the 'inverse Goldbach problem'. The answer is surely no. The best result in this direction is due to Elsholtz and Harper [ElHa15], who showed that if $A,B$ are such sets then for all large $x$ we must have $\frac{x^{1/2}}{\log x\log\log x} \ll \lvert A \cap [1,x]\rvert \ll x^{1/2}\log\log x$ and similarly for $B$.

Elsholtz [El01] has proved there are no infinite sets $A,B,C$ such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.

Let $A,B\subseteq \mathbb{N}$ be two infinite sets. How dense can $A+B$ be if all elements of $A+B$ are pairwise relatively prime?
Asked by Straus, inspired by a problem of Ostmann (see ).
If $A\subset \mathbb{N}$ is a finite set then let $G(A)$ denote the greatest integer which is not expressible as a finite sum of elements from $A$ (with repetitions allowed). Let $g(n,t)=\max G(A)$ where the maximum is taken over all $A\subseteq \{1,\ldots,t\}$ of size $\lvert A\rvert=n$ which has no common divisor. Is it true that $g(n,t)\sim \frac{t^2}{n-1}?$
This type of problem is associated with Frobenius. Erdős and Graham [ErGr72] proved $g(n,t)<2t^2/n$, and there are examples which show that $g(n,t) \geq \frac{t^2}{n-1}-5t$ for $n\geq 2$.
Let $k\leq n$. What choice of $A\subseteq \{1,\ldots,n\}$ of size $\lvert A\rvert=k$ maximises the number of integers not representable as the sum of finitely many elements from $A$ (with repetitions allowed)? Is it $\{n,n-1,\ldots,n-k+1\}$?
Associated with problems of Frobenius.
Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$. What is the largest integer not of the form $\sum_{1\leq i<n}c_i\binom{n}{i}$ where the $c_i\geq 0$ are integers?
If $p$ is a prime and $k,m\geq 2$ then let $r(k,m,p)$ be the minimal $r$ such that $r,r+1,\ldots,r+m-1$ are all $k$th power residues modulo $p$. Let $\Lambda(k,m)=\limsup_{p\to \infty} r(k,m,p).$ Is it true that $\Lambda(k,2)$ is finite for all $k$? Is $\Lambda(k,3)$ finite for all odd $k$? How large are they?
Asked by Lehmer and Lehmer [LeLe62]. For example $\Lambda(2,2)=9$ and $\Lambda(3,2)=77$. It is known that $\Lambda(k,3)=\infty$ for all even $k$ and $\Lambda(k,4)=\infty$ for all $k$.

This has been partially resolved: Hildebrand [Hi91] has shown that $\Lambda(k,2)$ is finite for all $k$.

Let $1\leq a_1<\cdots<a_k\leq x$. How many of the partial products $a_1,a_1a_2,\ldots,a_1\cdots a_k$ can be squares? Is it true that, for any $\epsilon>0$, there can be more than $x^{1-\epsilon}$ squares?
There are trivially $o(x)$ many squares.
How large can $A\subseteq \{1,\ldots,N\}$ be if $A+A$ contains no square numbers?
Taking all integers $\equiv 1\pmod{3}$ shows that $\lvert A\rvert\geq N/3$ is possible. This can be improved to $\tfrac{11}{32}N$ by taking all integers $\equiv 1,5,9,13,14,17,21,25,26,29,30\pmod{32}$, as observed by Massias.

Lagarias, Odlyzko, and Shearer [LOS83] proved this is sharp for the modular version of the problem; that is, if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ is such that $A+A$ contains no squares then $\lvert A\rvert\leq \tfrac{11}{32}N$. They also prove the general upper bound of $\lvert A\rvert\leq 0.475N$ for the integer problem.

In fact $\frac{11}{32}$ is sharp in general, as shown by Khalfalah, Lodha, and Szemerédi [KLS02], who proved that the maximal such $A$ satisfies $\lvert A\rvert\leq (\tfrac{11}{32}+o(1))N$.

Let $G$ be the infinite graph on $\mathbb{N}$ where we connect $m,n$ by an edge if and only if $n+m$ is a square. Is the chromatic number of $G$ equal to $\aleph_0$? What if $n+m$ is required to be a $k$th power?
Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite and let $A(x)$ count the numer of indices for which $\mathrm{lcm}(a_i,a_{i+1})\leq x$. Is it true that $A(x) \ll x^{1/2}$? How large can $\liminf \frac{A(x)}{x^{1/2}}$ be?
It is easy to give a sequence with $\limsup\frac{A(x)}{x^{1/2}}=c>0.$ There are related results (particularly for what happens if we consider $\mathrm{lcm}(a_i,a_{i+1},\ldots,a_{i+k})$ in a paper of Erdős and Szemerédi [ErSz80].
Let $N\geq 1$. What is the size of the largest $A\subset \{1,\ldots,N\}$ such that $\mathrm{lcm}(a,b)\leq N$ for all $a,b\in A$? Is it attained by choosing all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$?
Is it true that if $A\subseteq\mathbb{N}$ is such that $\frac{1}{\log\log x}\sum_{n\in A\cap [1,x)}\frac{1}{n}\to \infty$ then $\left(\sum_{n\in A\cap [1,x)}\frac{1}{n}\right)^{-2} \sum_{\substack{a,b\in A\cap (1,x]\\ a<b}}\frac{1}{\mathrm{lcm}(a,b)}\to \infty?$
Let $m,n\geq 1$. What is $\# \{ k(m-k) : 1\leq k\leq m/2\} \cap \{ l(n-l) : 1\leq l\leq n/2\}?$ Can it be arbitrarily large? Is it $\leq (mn)^{o(1)}$ for all suffiicently large $m,n$?
Let $A\subseteq\mathbb{N}$ be infinite and $d_A(n)$ count the number of $a\in A$ which divide $n$. Is it true that, for every $k$, $\limsup_{x\to \infty} \frac{\max_{n<x}d_A(n)}{\left(\sum_{n\in A\cap[1,x)}\frac{1}{n}\right)^k}=\infty?$
The answer is yes, proved by Erdős and Sárkőzy [ErSa80].
Is it true that, for any $c>1/2$, if $p$ is a large prime and $n$ is sufficiently large (both depending on $c$) then there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
An unpublished result of Heilbronn guarantees this for $c$ sufficiently close to $1$.
Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$. What is the growth rate of $\delta(n)$?

If $\delta'(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta'(n)=o(\delta(n))$?

Erdős [Er35] proved that $\delta(n)<(\log n)^{-c}$ for some $c>0$. Tenenbaum [Te75] showed that $\delta(n)=(\log n)^{(-1+o(1))\alpha}$ where $\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.$

This has been resolved by Ford [Fo08]. Among many other results, Ford proves $\delta(n)\asymp \frac{1}{(\log n)^\alpha(\log\log n)^{3/2}},$ and that the second conjecture is false (i.e. $\delta'(n) \geq \delta(n)$ for some constant $c>0$)

Let $\tau(n)$ count the divisors of $n$ and $\tau^*(n)$ count the number of $k$ such that $n$ has a divisor in $[2^k,2^{k+1})$. Is it true that, for all $\epsilon>0$, $\tau^*(n) < \epsilon \tau(n)$ for almost all $n$?
Erdős and Graham also ask whether there is a good inequality known for $\sum_{n\leq x}\tau^*(n)$.
Let $r(n)$ count the number of $d_1,d_2$ such that $d_1\mid n$ and $d_2\mid n$ and $d_1<d_2<2d_1$. Is it true that, for every $\epsilon>0$, $r(n) < \epsilon \tau(n)$ for almost all $n$, where $\tau(n)$ is the number of divisors of $n$?
How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
Estimate $n_k$, the smallest integer such that $\prod_{1\leq i\leq k}(n_k-i)$ has no prime factor in $(k,2k)$.
Erdős and Graham write 'we can prove $n_k>k^{1+c}$ but no doubt much more is true'.
Let $\omega(n)$ count the number of distinct prime factors of $n$. What is the size of the largest interval $I\subseteq [x,2x]$ such that $\omega(n)>\log\log n$ for all $n\in I$?
Erdős [Er37] proved that the density of integers $n$ with $\omega(n)>\log\log n$ is $1/2$. The Chinese remainder theorem implies that there is such an interval with $\lvert I\rvert \geq (1+o(1))\frac{\log x}{(\log\log x)^2}.$ It could be true that there is such an interval of length $(\log x)^{k}$ for arbitrarily large $k$.
Is it true that, for all sufficiently large $n$, $p_n^2 \leq p_{n+i}p_{n-i}$ for all $i<n$, where $p_k$ is the $k$th prime?
Asked by Erdős and Straus. The answer is no, as shown by Pomerance [Po79].
Let $f(n) = \min_{i<n} (p_{n+i}+p_{n-i}),$ where $p_k$ is the $k$th prime. Is it true that $\limsup_n (f(n)-2p_n)=\infty?$
Pomerance [Po79] has proved the $\limsup$ is at least $2$.
Let $q_1<q_2<\cdots$ be a sequence of primes such that $q_{n+1}-q_n\geq q_n-q_{n-1}.$ Must $\lim_n \frac{q_n}{n^2}=\infty?$
Richter [Ri76] proved that $\liminf_n \frac{q_n}{n^2}>2.84\cdots.$
Let $s_n$ be the smallest prime such that $n\mid s_n-1$ and let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$. Is it true that $s_n>m_n$ for almost all $n$? Does $s_n/m_n\to \infty$ for almost all $n$? Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?
Is there some $\epsilon>0$ such that there are infinitely many $n$ where all primes $p\leq (2+\epsilon)\log n$ divide $\prod_{1\leq i\leq \log n}(n+i)?$
Let $A_n$ be the least common multiply of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$, $A_{p_{k+1}-1}< p_kA_{p_k}?$
Erdős and Graham write this is 'almost certainly' true, but the proof is beyond our ability, for two reasons (at least):
• Firstly, one has to rule out the possibility of many primes $q$ such that $p_k<q^2<p_{k+1}$. There should be at most one such $q$, which would follow from $p_{k+1}-p_k<p_k^{1/2}$, which is essentially the notorious Legendre's conjecture.
• The small primes also cause trouble.
Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$. Estimate $f(u)$.
Let $a_0=n$ and $a_1=1$, and in general $a_k$ is the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $1\leq i<k$. Does $\sum_{i}\frac{1}{a_i}\to \infty$ as $n\to \infty$? What about if we restrict the sum to those $i$ such that $n-a_j$ is divisible by some prime $\leq a_j$, or the complement of such $i$?
This question arose in work of Eggleton, Erdős, and Selfridge.
Let $s_t(n)$ be the $t$-smooth component of $n$ - that is, the product of all primes $p$ (with multiplicity) dividing $n$ such that $p<t$. Let $f(n,t)$ count the number of distinct possible values for $s_t(m)$ for $m\in [n+1,n+t]$. Is there an $\epsilon>0$ such that $f(n,t)>\epsilon t$ for all $t$ and $n$?
Erdős and Graham report they can show $f(n,t) \gg \frac{t}{\log t}.$
Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that $\sum_{\substack{n<x\\ n\textrm{ not prime}}}\frac{p(n)}{n}\sim c\frac{x^{1/2}}{(\log x)^2}.$ Is it true that $\sum_{x\leq n\leq x+cx^{1/2}(\log x)^2}\frac{p(n)}{n} \gg 1$ for all large $x$?
Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that $n+f(n)<m<n+p(m)?$ (Here $p(m)$ is the least prime factor of $m$.)
Let $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ be a lacunary sequence (so there exists some $\lambda>1$ with $a_{k+1}\geq \lambda a_k$ for all $k$). Is there an irrational $\alpha$ such that $\{ \{\alpha a_k\} : k\geq 1\}$ is not everywhere dense in $[0,1]$ (where $\{x\}=x-\lfloor x\rfloor$ is the fractional part).
Erdős and Graham write the existence of such an $\alpha$ has 'very recently been shown', but frustratingly give neither a name nor a reference. I will try to track down this solution soon.
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that $\| \lvert P_i-P_j\rvert \| \geq \delta$ for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is it true that, for any $0<\delta<1/2$, we have $N(X,\delta)=o(X)?$ In fact, is it true that (for any fixed $\delta>0$) $N(X,\delta)<X^{1/2+o(1)}?$

The first conjecture was proved by Sárközy [Sa76], who in fact proved $N(X,\delta) \ll \delta^{-3}\frac{X}{\log\log X}.$ It is not even known whether $N(X,\delta)<X^{1-\epsilon}$ for some $\epsilon>0$. See also .
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that $\| \lvert P_i-P_j\rvert \| \geq \delta$ for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)

Is there some $\delta>0$ such that $\lim_{x\to \infty}N(X,\delta)=\infty?$

Graham proved this is true, and in fact $N(X,1/10)> \frac{\log X}{10}.$ This was substantially improved by Sárközy [Sa76], who proved that for, all sufficiently small $\delta>0$, $N(X,\delta)>X^{1/2-\delta^{1/7}}.$ See also .
Prove the following for all large $x$: there is a choice of congruence classes $a_p$ for all primes $p\leq x$, and a decomposition $\{p\leq x\}=A\sqcup B$ into two non-empty sets such that, for all $n<x$, there exist some $p\in A$ and $q\in B$ such that $n\equiv a_p\pmod{p}$ and $n\equiv a_q\pmod{q}$.
This is what I assume the intended problem is, although the presentation in [ErGr80] is missing some crucial quantifiers, so I may have misinterpreted it.
For any $n$ let $D_n$ be the set of sums of the shape $d_1,d_1+d_2,d_1+d_2+d_3,\ldots$ where $1<d_1<d_2<\cdots$ are the divisors of $n$.

What is the size of $D_n\backslash \cup_{m<n}D_m$?

If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?

Let $A$ be the set of all $n$ such that $n=d_1+\cdots+d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m\mid n$. Does $\sum_{n\in A}\frac{1}{n}$ converge?
The same question can be asked for those $n$ which do not have distinct sums of sets of divisors, but any proper divisor of $n$ does.
Call $n$ weird if $\sigma(n)\geq 2n$ and $n\neq d_1+\cdots+d_k$, where the $d_i$ are distinct proper divisors of $n$. Are there any odd weird numbers? Are there infinitely many primitive weird numbers, i.e. those such that no proper divisor of $n$ is weird?
Weird numbers were investigated by Benkoski and Erdős [BeEr74], who proved that the set of weird numbers has positive density.

Melfi [Me15] has proved that there are infinitely many primitive weird numbers, conditional on various well-known conjectures on the distribution of prime gaps. For example, it would suffice to show that $p_{n+1}-p_n <\frac{1}{10}p_n^{1/2}$ for sufficiently large $n$.

Given a finite set of primes $Q=Q_0$, define a sequence of sets $Q_i$ by letting $Q_{i+1}$ be $Q_i$ together with all primes formed by adding three distinct elements of $Q_i$. Is there some initial choice of $Q$ such that the $Q_i$ become arbitrarily large?
A problem of Ulam. In particular, what about $Q=\{3,5,7,11\}$?
Given some initial finite sequence of primes $q_1<\cdots<q_m$ extend it so that $q_{n+1}$ is the smallest prime of the form $q_n+q_i-1$ for $n\geq m$. Is there an initial starting sequence so that the resulting sequence is infinite?
A problem due to Ulam. For example if we begin with $3,5$ then the sequence continues $3,5,7,11,13,17,\ldots$. It is possible that this sequence is infinite.
Is there a permutation $a_1,a_2,\ldots$ of the positive integers such that $a_k+a_{k+1}$ is always prime?

Watts has suggested that perhaps the obvious greedy algorithm defines such a permutation - that is, let $a_1=1$ and let $a_{n+1}=\min \{ x : a_n+x\textrm{ is prime and }x\neq a_i\textrm{ for }i\leq n\}.$ In other words, do all positive integers occur as some such $a_n$? Do all primes occur as a sum?

Let $b_1=1$ and in general let $b_{n+1}$ be the least integer which is not of the shape $\sum_{u\leq i\leq v}b_i$ for some $1\leq u\leq v\leq n$. How does this sequence grow?
The sequence is OEIS A002048 and begins $1,2,4,5,8,10,14,15,16,21,22,23,25,\ldots.$ In general, if $a_1<a_2<\cdots$ is a sequence so that no $a_n$ is a sum of consecutive $a_i$s, then must the density of the $a_i$ be zero? What about the lower density?
Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$, is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct?
A problem of Graham.
Let $A\subseteq \mathbb{F}_p$. Let $A\hat{+}A = \{ a+b : a\neq b \in A\}.$ Is it true that $\lvert A\hat{+}A\rvert \geq \min(2\lvert A\rvert-3,p)?$
A question of Erdős and Heilbronn. Solved in the affirmative by da Silva and Hamidoune [dSHa94].
Let $f:\mathbb{Z}\to \mathbb{Z}$ be a polynomial of degree at least $2$. Is there a set $A$ such that every $z\in \mathbb{Z}$ has exactly one representation as $z=a+f(n)$ for some $a\in A$ and $n\in \mathbb{Z}$?
Probably there is no such $A$ for any polynomial $f$.
Let $p$ be a prime and $A_p = \{ k! \mod{p} : 1\leq k<p\}.$ Is it true that $\lvert S_p\rvert \sim (1-\tfrac{1}{e})p?$
Is it true that, for all $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?
A conjecture of Graham. It is easy to see that $2^n\not\equiv 1\mod{n}$ for all $n>1$, so the restriction $k\neq 1$ is necessary. Erdős and Graham report that Graham, Lehmer, and Lehmer have proved this for $k=2^i$ for $i\geq 1$, or if $k=-1$, but I cannot find such a paper.

As an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\equiv 3\pmod{n}$ is $n=4700063497$.

Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence. Is it true that there are infinitely many $m,n$ such that $\lvert x_{m+n}-x_n\rvert \leq \frac{1}{\sqrt{5}n}?$
A conjecture of Newman. This was proved Chung and Graham, who in fact show that for any $\epsilon>0$ there must exist some $n$ such that there are infinitely many $m$ for which $\lvert x_{m+n}-x_m\rvert < \frac{1}{(c-\epsilon)n}$ where $c=1+\sum_{k\geq 1}\frac{1}{F_{2k}}=2.535\cdots$ and $F_m$ is the $m$th Fibonacci number. This constant is best possible.
Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$. For any $A\subset \mathbb{N}$ let $T(A) = \{ a_ix+b_i : 1\leq i\leq r\textrm{ and }x \in A\}.$ Prove that, if $A_1=\{1\}$ and $A_{i+1}=T(A_i)$, then $\min(A_i) \ll 1$ for all $i\geq 0$.
Erdős and Graham write that 'it is surprising that [this problem] offers difficulty'.
Define a sequence by $a_1=1$ and $a_{n+1}=\lfloor\sqrt{2}(a_n+1/2)\rfloor$ for $n\geq 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in the binary expansion of $\sqrt{2}$.

Find similar results for $\theta=\sqrt{m}$, and other algebraic numbers.

The result for $\sqrt{2}$ was obtained by Graham and Pollak [GrPo70]. The problem statement is open-ended, but presumably Erdős and Graham would have been satisfied with the wide-ranging generalisations of Stoll ([St05] and [St06]).
Let $f(k)$ be the minimal $N$ such that if $\{1,\ldots,N\}$ is $k$-coloured then there is a monochromatic solution to $a+b=c$. Estimate $f(k)$. In particular, is it true that $f(k) < c^k$ for some constant $c>0$?
Schur proved that $f(k)<ek!$. See also .
Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class).
A conjecture of Roth.
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let $B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\}.$ Must $B$ have a logarithmic density, i.e. is it true that $\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}$ exists?
Davenport and Erdős [DaEr37] proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. The problem considers logarithmic density since Besicovitch [Be34] showed examples exist without a natural density, even when $X_n=\{0\}$ for all $n\in A$.
Let $A\subseteq \mathbb{N}$ have positive density. Must there exist distinct $a,b,c\in A$ such that $[a,b]=c$ (where $[a,b]$ is the lowest common multiple of $a$ and $b$)?
Davenport and Erdős [DaEr37] showed that there must exist an infinite sequence $a_1<a_2\cdots$ in $A$ such that $a_i\mid a_j$ for all $i\leq j$.

This is true, a consequence of the positive solution to  by Kleitman [Kl71].

Let $A$ be a finite set and $B=\{ n \geq 1 : a\nmid n\textrm{ for all }a\in A\}.$ Is it true that, for every $m>n$, $\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$
The example $A=\{a\}$ and $n=2a-1$ and $m=2a$ shows that $2$ would be best possible.
Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}.$ If $B=\{b_1<b_2<\cdots\}$ then is it true that $\lim \frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2$ exists (and is finite)?
For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős.

Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that $\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?$
This would be best possible, for example letting $A=[1,N/2]\cap \mathbb{N}$ and $B=\{ N/2<p\leq N: p\textrm{ prime}\}$.

Let $f:\mathbb{N}\to \mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). If there is a constant $c$ such that $\lvert f(n+1)-f(n)\rvert <c$ for all $n$ then must there exist some $c'$ such that $f(n)=c'\log n+O(1)?$
Erdős [Er46] proved that if $f(n+1)-f(n)=o(1)$ or $f(n+1)\geq f(n)$ then $f(n)=c\log n$ for some constant $c$.

This is true, and was proved by Wirsing [Wi70].

Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ be infinite such that $a_{i+1}/a_i\to 1$. For any $x\geq a_1$ let $f(x) = \frac{x-a_i}{a_{i+1}-a_i}\in [0,1),$ where $x\in [a_i,a_{i+1})$. Is it true that, for almost all $\alpha$, the sequence $f(\alpha n)$ is uniformly distributed in $[0,1)$?
For example if $A=\mathbb{N}$ then $f(x)=\{x\}$ is the usual fractional part operator.

A problem due to Le Veque [LV53], who proved it in some special cases.

This is false is general, as shown by Schmidt [Sc69].

Does there exist a $k$ such that every sufficiently large integer can be written in the form $\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$ for some integers $a_i\geq 2$?
A conjecture of Schinzel.
Let $\alpha,\beta \in \mathbb{R}$. Is it true that $\liminf_{n\to \infty} n \| n\alpha \| \| n\beta\| =0$ where $\|x\|$ is the distance from $x$ to the nearest integer?
The infamous Littlewood conjecture.
Let $\alpha \in \mathbb{R}$ be irrational and $\epsilon>0$. Are there integers $x,y,z$ such that $\lvert x^2+y^2-z^2\alpha\rvert <\epsilon?$
Originally a conjecture due to Oppenheim. Davenport and Heilbronn [DaHe46] solve the analogous problem for quadratic forms in 5 variables.

This is true, and was proved by Margulis [Ma89].