Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown (contrary to Erdős' expectations) that the answer is no: the smallest modulus must be at most $10^{18}$.

An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the upper bound on the smallest modulus to $616000$.

The best known lower bound is a covering system whose minimum modulus is $42$, due to Owens [Ow14].

Asked by Erdős and Selfridge (sometimes also with Schinzel). They also asked whether there can be a covering system such that all the moduli are odd and squarefree. The answer to this stronger question is no, proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

Hough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

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 $n_j$ (but the latter has been shown not to exist, see [586]).

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.

Crocker [Cr71] has proved there are are $\gg\log\log N$ such integers in $\{1,\ldots,N\}$. Pan [Pa11] improved this to $\gg_\epsilon N^{1-\epsilon}$ for any $\epsilon>0$. Erdős believed this cannot be proved by covering systems, i.e. integers of the form $p+2^k+2^l$ exist in every infinite arithmetic progression.

The sequence of such numbers is A006286 in the OEIS.

See also [10], [11], and [16].

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 $1-\epsilon$.

Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).

See also [9], [11], and [16].

Odlyzko has checked this up to $10^7$. Hercher [He24b] has verified this is true for all odd integers up to $2^{50}\approx 1.12\times 10^{15}$.

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 often asked this under the weaker assumption that $n$ is not divisible by $4$. Erdős thought that proving this with two powers of 2 is perhaps easy, and could prove that it is true (with a single power of two) for almost all $n$.

See also [9], [10], and [16].

Asked by Erdős and Sárközy [ErSa70], who proved that $A$ must have density $0$. They also prove that this is essentially best possible, in that given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with this property and infinitely many $N$ such that \[\lvert A\cap\{1,\ldots,N\}\rvert>\frac{N}{f(N)}.\] (Their example is given by all integers in $(y_i,\frac{3}{2}y_i)$ congruent to $1$ modulo $(2y_{i-1})!$, where $y_i$ is some sufficiently quickly growing sequence.)

An example of an $A$ with this property where \[\liminf \frac{\lvert A\cap\{1,\ldots,N\}\rvert}{N^{1/2}}\log N>0\] is given by the set of $p^2$, where $p\equiv 3\pmod{4}$ is prime.

Elsholtz and Planitzer [ElPl17] have constructed such an $A$ with \[\lvert A\cap\{1,\ldots,N\}\rvert\gg \frac{N^{1/2}}{(\log N)^{1/2}(\log\log N)^2(\log\log\log N)^2}.\]

Schoen [Sc01] proved that if all elements in $A$ are pairwise coprime then \[\lvert A\cap\{1,\ldots,N\}\rvert \ll N^{2/3}\] for infinitely many $N$. Baier [Ba04] has improved this to $\ll N^{2/3}/\log N$.

For the finite version see [13].

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].

For the infinite version see [12].

In [Er92c] Erdős asks about the general version where $a\nmid (b_1+\cdots+b_r)$ for $a<\min(b_1,\ldots,b_r)$, and whether $\lvert A\rvert \leq N/(r+1)+O(1)$.

Apparently originally considered by Erdős and Nathanson, although later Erdős attributes this to Erdős, Sárközy, and Szemerédi (but gives no reference), and claims a construction of an $A$ such that for all $\epsilon>0$ and all large $N$ \[\lvert \{1,\ldots,N\}\backslash B\rvert \ll_\epsilon N^{1/2+\epsilon},\] and yet there for all $\epsilon>0$ there exist infinitely many $N$ where \[\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/3-\epsilon}.\]

Erdös and Freud investigated the finite analogue in 'a recent Hungarian paper', proving that there exists $A\subseteq \{1,\ldots,N\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.

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.

In [Er98] Erdős further conjectures that \[\sum_{n=1}^\infty (-1)^n \frac{1}{n(p_{n+1}-p_n)}\] converges and \[\sum_{n=1}^\infty (-1)^n \frac{1}{p_{n+1}-p_n}\] diverges. He further conjectures that \[\sum_{n=1}^\infty (-1)^n \frac{1}{n(p_{n+1}-p_n)(\log\log n)^c}\] converges for every $c>0$, and reports that he and Nathanson can prove this for $c>2$ (and conditionally for $c=2$).

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 \[\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\] for some constant $c>0$. 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 $1558/1167=1.33504\cdots$).

There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Rudnev and Stevens above. The best bound for complex numbers is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}},\] due to Solymosi [So05].

One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that if $A\subseteq \mathbb{F}_p$ with $\lvert A\rvert <p^{5/8}$ then \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{11}{9}+o(1)},\] due to Rudnev, Shakan, and Shkredov [RSS20].

There is also a natural generalisation to higher-fold sum and product sets. For example, in [ErSz83] (and in [Er91]) Erdős and Szemerédi also conjecture that for any $m\geq 2$ and finite set of integers $A$ \[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\] See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. See also [818] for a special case.

Asked by Erdős and Szemerédi [ErSz83]. Solved in this form by Chang [Ch03].

Erdős and Szemerédi proved that there exist arbitrarily large sets $A$ such that the integers which are the sum or product of distinct elements of $A$ is at most \[\exp\left(c (\log \lvert A\rvert)^2\log\log\lvert A\rvert\right)\] for some constant $c>0$.

See also [52].

Conjectured by Erdős, Sós, and Sárkzözy [ESS95], who proved \[F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N,\] \[F_3(N) = (1-o(1))N,\] and also established asymptotics for $F_k(N)$ for all even $k\geq 4$ (in particular $F_k(N)\asymp N/\log N$ for all even $k\geq 4$). Erdős [Er38] earlier proved that $F_4(N)=o(N)$ - indeed, if $\lvert A\rvert \gg N$ and $A\subseteq \{1,\ldots,N\}$ then there is a non-trivial solution to $ab=cd$ with $a,b,c,d\in A$.

Erdős (and independently Hall [Ha96] and Montgomery) also asked about $F(N)$, the size of the largest $A\subseteq\{1,\ldots,N\}$ such that the product of no odd number of $a\in A$ is a square. Ruzsa [Ru77] observed that $1/2<\lim F(N)/N <1$. Granville and Soundararajan [GrSo01] proved an asymptotic \[F(N)=(1-c+o(1))N\] where $c=0.1715\ldots$ is an explicit constant.

This problem was answered in the negative by Tao [Ta24], who proved that for any $k\geq 4$ there is some constant $c_k>0$ such that $F_k(N) \leq (1-c_k+o(1))N$.

See also [888].

Conjectured by Erdős, Pomerance, and Sárközy [ErPoSa97] who prove this when $f$ is the divisor function or the number of distinct prime divisors of $n$, but Erdős believed it is false when $f(n)=\phi(n)$ or $\sigma(n)$.

Conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.

In [Er92b] Erdős wrote 'last year I made the following silly conjecture': every integer $n$ can be written as the sum of distinct integers of the form $2^k3^l$, none of which divide any other. 'I mistakenly thought that this was a nice and difficult conjecture but Jansen and several others found a simple proof by induction.' This simple proof is as follows: one proves the stronger fact that such a representation always exists, and moreover if $n$ is even then all the summands can be taken to be even: if $n=2m$ we are done applying the inductive hypothesis to $m$. Otherwise if $n$ is odd then let $3^k$ be the largest power of $3$ which is $\leq n$ and apply the inductive hypothesis to $n-3^k$ (which is even).

In [Er92b] Erdős makes the stronger conjecture (for $a=2$, $b=3$, and $c=5$) that, for any $\epsilon>0$, all large integers $n$ can be written as the sum of distinct integers $b_1<\cdots <b_t$ of the form $2^k3^l5^m$ where $b_t<(1+\epsilon)b_1$.

See also [845].

Conjectured by Burr, Erdős, Graham, and Li [BEGL96]. Pomerance observed that the condition $\sum 1/(d_i-1)\geq 1$ is necessary. In [BEGL96] they prove the property holds for $\{3,4,7\}$.

See also [125].

A problem of Burr, Erdős, Graham, and Li [BEGL96]. More generally, if $n_1<\cdots<n_k$ have \[\sum_{i=1}^k\log_{n_k}(2)>1\] and $A_i$ is the set of integers with only the digits $0,1$ in base $n_i$ then does $A_1+\cdots+A_k$ have positive density?

See also [124].

Investigated by Erdős and Turán [ErTu34] (prompted by a question of Lázár and Grünwald) 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.

Asked by Schinzel, motivated by a question of Erdős and Selfridge (see [7]). The answer is no, as proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].

A problem of Erdős and Szemerédi, which strengthens the conjecture [52].

This strong conjecture was disproved by Alon, Ruzsa, and Solymosi [ARS20], who constructed (for arbitrarily large $n$) a set of integers $A$ with $\lvert A\rvert=n$ and a graph $G$ with $\gg n^{5/3-o(1)}$ many edges such that \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \ll \lvert A\rvert^{4/3+o(1)}.\] Alon, Ruzsa, and Solymosi do prove, however, that if $A$ has size $n$ and $G$ has $m$ edges then \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \gg m^{3/2}n^{-7/4}.\]