6 solved out of 17 shown (show only solved or open)
SOLVED - $100 Let$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 $d_n> d_{n+1}>\cdots >d_{n+m}.$

OPEN
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 $h(n!)<n^{o(1)}$? Or perhaps even $h(n!)<(\log n)^{O(1)}$?

It is easy to see that almost all numbers are not practical. Erdős originally showed that $h(n!) <n$. Vose [Vo85] proved the existence of infinitely many practical $m$ such that $h(m)\ll (\log m)^{1/2}$.

Related to [304].

SOLVED - $250 The density of integers which have two divisors$d_1,d_2$such that$d_1<d_2<2d_1$exists and is equal to$1$. In [Er79] asks the stronger version with$2$replaced by any constant$c>1$. The answer is yes (also to this stronger version), proved by Maier and Tenenbaum [MaTe84]. (Tenenbaum has told me that they received \$650 for their solution.)

OPEN
Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\alpha \geq 0$, $\lim_{x\to \infty}\frac{1}{x}\sum_{s_n\leq x}(s_{n+1}-s_n)^\alpha$ exists?
Erdős [Er51] proved this for all $0\leq \alpha \leq 2$, and Hooley [Ho73] extended this to all $\alpha \leq 3$.

SOLVED
Show that, for any $n\geq 5$, the binomial coefficient $\binom{2n}{n}$ is not squarefree.
It is easy to see that $4\mid \binom{2n}{n}$ except when $n=2^k$, and hence it suffices to prove this when $n$ is a power of $2$.

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 $f(n)\gg \log n$?

OPEN
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.

In [Er79] Erdős says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.

Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.

SOLVED - $250 Let$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$\gamma\geq 1$(which was also asked by Erdős in [Er73]). OPEN 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 $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ OPEN Is there some absolute constant$C>0$such that $\sum_{p\leq n}1_{p\nmid \binom{2n}{2}}\frac{1}{p}\leq C$ for all$n$? A question of Erdős, Graham, Ruzsa, and Straus [EGRS75], who proved that if$f(n)$is the sum in question then $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n) = \sum_{k=2}^\infty \frac{\log k}{2^k}=\gamma_0$ and $\lim_{x\to \infty}\frac{1}{x}\sum_{n\leq x}f(n)^2 = \gamma_0^2,$ so that for almost all integers$f(m)=\gamma_0+o(1)$. They also prove that, for all large$n$, $f(n) \leq c\log\log n$ for some constant$c<1$. (It is trivial from Mertens estimates that$f(n)\leq (1+o(1))\log\log n$.) A positive answer would imply that $\sum_{p\leq n}1_{p\mid \binom{2n}{n}}\frac{1}{p}=(1-o(1))\log\log n,$ and Erdős, Graham, Ruzsa, and Straus say there is 'no doubt' this latter claim is true. OPEN Let$\omega(n)$count the number of distinct primes dividing$n$. Are there infinitely many$n$such that, for all$m<n$, we have$m+\omega(m) \leq n$? Can one show that there exists an$\epsilon>0$such that there are infinitely many$n$where$m+\epsilon \omega(m)\leq n$for all$m<n$? In [Er79] Erdős calls such an$n$a 'barrier' for$\omega$. Some other natural number theoretic functions (such as$\phi$and$\sigma$) have no barriers because they increase too rapidly. Erdős believed that$\omega$should have infinitely many barriers. In [Er79d] he proves that$F(n)=\prod k_i$, where$n=\prod p_i^{k_i}$, has infinitely many barriers (in fact the set of barriers has positive density in the integers). Erdős also believed that$\Omega$, the count of the number of prime factors with multiplicity), should have infinitely many barriers. Selfridge found the largest barrier for$\Omega$which is$<10^5$is$99840$. In [ErGr80] this problem is suggested as a way of showing that the iterated behaviour of$n\mapsto n+\omega(n)$eventually settles into a single sequence, regardless of the starting value of$n$(see also [412] and [414]). Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'. See also [647] and [679]. SOLVED 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$n$? This is false, and was disproved by Erdős and Tenenbaum [ErTe81], who showed that in fact the upper density of the set of such$n$is$\asymp \epsilon^{1-o(1)}$(where the$o(1)$in the exponent$\to 0$as$\epsilon \to 0$). A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is$\ll\epsilon \log(2/\epsilon)$. Hall and Tenenbaum further prove that$\tau^+(n)/\tau(n)$has a distribution function. Erdős and Graham also asked whether there is a good inequality known for$\sum_{n\leq x}\tau^+(n)$. This was provided by Ford [Fo08] who proved $\sum_{n\leq x}\tau^+(n)\asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}}$ where $\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.$ See also [446] and [449]. Additional thanks to: Kevin Ford OPEN Let$\tau(n)$count the number of divisors of$n$. Is there some$n>24$such that $\max_{m<n}(m+\tau(m))\leq n+2?$ A problem of Erdős and Selfridge. This is true for$n=24$. The$n+2$is best possible here since $\max(\tau(n-1)+n-1,\tau(n-2)+n-2)\geq n+2.$ In [Er79] Erdős says 'it is extremely doubtful' that there are infinitely many such$n$, and in fact suggets that $\lim_{n\to \infty}\max_{m<n}(\tau(m)+m-n)=\infty.$ In [Er79d] Erdős says it 'seems certain' that for every$k$there are infinitely many$n$for which $\max_{n-k<m<n}(m+\tau(m))\leq n+2,$ but 'this is hopeless with our present methods', although it follows from Schinzel's Hypothesis H. See also [413]. SOLVED Are there any integer solutions to$x^xy^y=z^z$with$x,y,z>1$? Ko [Ko40] proved there are none if$(x,y)=1$, but there are in fact infinitely many solutions in general - for example, $x=2^{12}3^6, y = 2^83^8,\textrm{ and } z = 2^{11}3^7.$ More generally, writing$a=2^{n+1}$and$b=2^n-1$, $x = 2^{a(b-n)}b^{2b}\cdot 2^{2n},$ $y = 2^{a(b-n)}b^{2b}\cdot b^2,$ and $z = 2^{a(b-n)}b^{2b}\cdot 2^{n+1}b.$ In [Er79] Erdős asks if the infinite families found by Ko are the only solutions. OPEN We say that$A\subset \mathbb{N}$has the translation property if, for every$n$, there exists some integer$t_n\geq 1$such that $A\cap [1,n]=(A-t_n)\cap [1,n].$ • Does the set of the sums of two squares have the translation property? • If we partition all primes into$P\sqcup Q$, such that each set contains$\gg x/\log x$many primes$\leq x$for all large$x$, then can the set of integers only divisible by primes from$P$have the translation property? • If$A$is the set of squarefree numbers then how fast does the minimal such$t_n$grow? Is it true that$t_n>\exp(n^c)$for some constant$c>0$? Elementary sieve theory implies that the set of squarefree numbers has the translation property. More generally, Brun's sieve can be used to prove that if$B\subseteq \mathbb{N}$is a set of pairwise coprime integers with$\sum_{b<x}\frac{1}{b}=o(\log\log x)$then$A=\{ n: b\nmid n\textrm{ for all }b\in A\}$has the translation property. Erdős did not know what happens if the condition on$\sum_{b<x}\frac{1}{b}$is weakened or dropped altogether. OPEN Is every sufficiently large integer of the form $ap^2+b$ for some prime$p$and integer$a\geq 1$and$0\leq b<p$? The sieve of Eratosthenes implies that almost all integers are of this form, and the Brun-Selberg sieve implies the number of exceptions in$[1,x]$is$\ll x/(\log x)^c$for some constant$c>0$. Erdős [Er79] believed it is 'rather unlikely' that all large integers are of this form. What if the condition that$p$is prime is omitted? Selfridge and Wagstaff made a 'preliminary computer search' and suggested that there are infinitely many$n$not of this form even without the condition that$p$is prime. It should be true that the number of exceptions in$[1,x]$is$<x^c$for some constant$c<1$. Most generally, given some infinite set$A\subseteq \mathbb{N}$and function$f:A\to \mathbb{N}$one can ask for sufficient conditions on$A$and$f$that guarantee every large number (or almost all numbers) can be written as $am^2+b$ for some$m\in A$and$a\geq 1$and$0\leq b<f(m)$. In another direction, one can ask what is the minimal$c_n$such that$n$can be written as$n=ap^2+b$with$0\leq b<c_np$for some$p\leq \sqrt{n}$. This problem asks whether$c_n\leq 1$eventually, but in [Er79d] Erdős suggests that in fact$\limsup c_n=\infty$. Is it true that$c_n<n^{o(1)}$? OPEN Let$M(n,k)=[n+1,\ldots,n+k]$be the least common multiple of$\{n+1,\ldots,n+k\}$. Is it true that for all$m\geq n+k$$M(n,k) \neq M(m,k)?$ In general, how many solutions does$M(n,k)=M(m,l)$have when$m\geq n+k$and$l>1$? Erdős expects very few (and none when$l\geq k$). The only solutions Erdős knew were$M(4,3)=M(13,2)$and$M(3,4)=M(19,2)$. In [Er79d] Erdős conjectures the stronger fact that (aside from a finite number of exceptions) if$k>2$and$m\geq n+k$then$\prod_{i\leq k}(n+i)$and$\prod_{i\leq k}(m+i)$cannot have the same set of prime factors. See also [678] and [686]. OPEN Let$M(n,k)=[n+1,\ldots,n+k]$be the least common multiple of$\{n+1,\ldots,n+k\}$. Let$k\geq 3$. Are there infinitely many$m,n$with$m\geq n+k$such that $M(n,k)>M(m,k+1)?$ It is easy to see that there are infinitely many solutions to$M(n,k)>M(m,k)$. The referee of [Er79] found$M(96,7)>M(104,8)$and$M(132,7)>M(139,8)$. If$n_k$is the smallest$n$with this property (for some$m$) then are there good bounds for$n_k$? Erdős writes that he could prove$n_k/k\to \infty$, but knew of no good upper bounds. If$u_k$is minimal such that$M(u_k,k)>M(u_k+1,k)$and$t<\min(u_k,T)$then is it true that$M(t,k)\leq M(T,k)\$?