16 solved out of 29 shown (show only solved or open)
OPEN - $500 If$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert=n$is such that the subset sums$\sum_{a\in S}a$are distinct for all$S\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$[0,Nn)$. 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of the constant have been given (see [St23] for a history), with the current record$\sqrt{2/\pi}$first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound$N\geq \binom{n}{\lfloor n/2\rfloor}$. In [Er73] and [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. (The second proof of [DFX21] applies also to this generalisation.) This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink". See also [350]. Additional thanks to: Zachary Hunter SOLVED -$1000
Can the smallest modulus of a covering system be arbitrarily large?
Described by Erdős as 'perhaps my favourite problem'. Hough [Ho15], building on work of Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], has shown 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 bound on the smallest modulus to $616000$.

SOLVED - $10000 For any$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], who proved there exists some constant$C>0$such that the claim holds. Solved by Maynard [Ma16] and Ford, Green, Konyagin, and Tao [FGKT16]. The best bound available, due to all five authors [FGKMT18], 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$c>0$. OPEN 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. 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 $d_n> d_{n+1}>\cdots >d_{n+m}.$ Additional thanks to: Mehtaab Sawhney OPEN Is there a covering system all of whose moduli are odd? 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]). Additional thanks to: Antonio Girao OPEN -$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 the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.

Erdős and Sárközy conjectured the stronger version that if $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.

See also [40].

OPEN
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 $a\in A$? Can the bound $O(\log N)$ be achieved?
Erdős [Er54] proved that such a set $A$ exists with $\lvert A\cap\{1,\ldots,N\}\rvert\ll (\log N)^2$ (improving a previous result of Lorentz [Lo54] who achieved $\ll (\log N)^3$). Wolke [Wo96] has shown that such a bound is almost true, in that we can achieve $\ll (\log N)^{1+o(1)}$ if we only ask for almost all integers to be representable.
SOLVED - $500 If$f:\mathbb{N}\to \{-1,+1\}$then is it true that for every$C>0$there exist$d,m\geq 1$such that $\left\lvert \sum_{1\leq k\leq m}f(kd)\right\rvert > C?$ The Erdős discrepancy problem. This is true, and was proved by Tao [Ta16], who also proved the more general case when$f$takes values on the unit sphere. OPEN -$500
Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?
A $\sqrt{n}\times\sqrt{n}$ integer grid shows that this would be the best possible. Nearly solved by Guth and Katz [GuKa15] who proved that there are always $\gg n/\log n$ many distinct distances.

A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points.

See also [661].

SOLVED
If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances.
Solved by Altman [Al63]. The stronger variant that says there is one point which determines at least $\lfloor \frac{n+1}{2}\rfloor$ distinct distances is still open. Fishburn in fact conjectures that if $R(x)$ counts the number of distinct distances from $x$ then $\sum_{x\in A}R(x) \geq \binom{n}{2}.$

Szemerédi conjectured (see [Er97e]) that this stronger variant remains true if we only assume that no three points are on a line, and proved this with the weaker bound of $n/3$.

See also [660].

SOLVED - $100 Let$z_i$be an infinite sequence of complex numbers such that$\lvert z_i\rvert=1$for all$i\geq 1$, and for$n\geq 1$let $p_n(z)=\prod_{i\leq n} (z-z_i).$ Let$M_n=\max_{\lvert z\rvert=1}\lvert p_n(z)\rvert$. Is it true that$\limsup M_n=\infty$? Is it true that there exists$c>0$such that for infinitely many$n$we have$M_n > n^c$, or even that for all$n$$\sum_{k\leq n}M_k > n^{1+c}?$ The weaker conjecture that$\limsup M_n=\infty$was proved by Wagner, who show that there is some$c>0$with$M_n>(\log n)^c$infinitely often. This was solved by Beck [Be91], who proved that there exists some$c>0$such that $\max_{n\leq N} M_n > N^c.$ Additional thanks to: Winston Heap OPEN Let the van der Waerden number$W(k)$be such that whenever$N\geq W(k)$and$\{1,\ldots,N\}$is$2$-coloured there must exist a monochromatic$k$-term arithmetic progression. Improve the bounds for$W(k)$- for example, prove that$W(k)^{1/k}\to \infty$. When$p$is prime Berlekamp [Be68] has proved$W(p+1)\geq p2^p$. Gowers [Go01] has proved $W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}}.$ SOLVED -$1000
Let $r_k(N)$ be the size of the largest subset of $\{1,\ldots,N\}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N)=o(N)$.
Proved by Szemerédi [Sz74]. The best known bounds are due to Kelley and Meka [KeMe23] for $k=3$, Green and Tao [GrTa17] for $k=4$, and Leng, Sah, and Sawhney [LSS24] for $k\geq 5$.

See also [3].

Additional thanks to: Zachary Chase
OPEN
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 $d_{n+1}=d_n$.
SOLVED
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.
SOLVED
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 $\gamma\geq 1$.
SOLVED
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] proved there is such a set with, for all large $N$, $\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N$ The answer is yes, proved by Ruzsa [Ru72]. Ruzsa's construction is ingeniously simple: $A = \{ 5^nm : m\geq 1\textrm{ and }5^n\geq C\log m\}+\{0,1\}$ for some large absolute constant $C>0$. That every large integer is of the form $2^k+a$ for some $a\in A$ is a consequence of the fact that $2$ is a primitive root of $5^n$ for all $n\geq 1$.
OPEN
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 $n_{k+1}-n_k \ll n_k^{1/4}.$
OPEN
Let $d\geq 2$ and $n\geq 2$. Let $f_d(n)$ be maximal such that, for any $A\subseteq \mathbb{R}^d$ of size $n$, with diameter $1$, the distance 1 occurs between $f_d(n)$ many pairs of points in $A$. Estimate $f_d(n)$.
Hopf and Pannwitz [HoPa34] proved $f_2(n)=n$. Heppes [He56] and Grünbaum-Strasziewicz independently showed that $f_3(n)=2n-2$.
SOLVED
If $A\subseteq \mathbb{R}^d$ is any set of $2^d+1$ points then some three points in $A$ determine an obtuse angle.
For $d=2$ this is trivial. For $d=3$ there is an unpublished proof by Kuiper and Boerdijk. The general case was proved by Danzer and Grünbaum [DaGr62].
Additional thanks to: Boris Alexeev and Dustin Mixon
SOLVED
Let $f(\theta) = \sum_{k\geq 1}c_k e^{ik\theta}$ be a trigonometric polynomial (so that the $c_k\in \mathbb{C}$ are finitely supported) with real roots such that $\max_{\theta\in [0,2\pi]}\lvert f(\theta)\rvert=1$. Then $\int_0^{2\pi}\lvert f(\theta)\rvert \mathrm{d}\theta \leq 4.$
This was solved independently by Kristiansen [Kr74] (only in the case when $c_k\in\mathbb{R}$) and Saff and Sheil-Small [SSS73] (for general $c_k\in \mathbb{C}$).
Additional thanks to: Winston Heap, Vjekoslav Kovac, Karlo Lelas
OPEN
Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?
More generally, if $A,B\subseteq \mathbb{R}$ are two countable dense sets then is there an entire function such that $f(A)=B$?
Additional thanks to: Boris Alexeev and Dustin Mixon
SOLVED
Let $f=\sum_{n=0}^\infty a_nz^n$ be an entire function. Is it true that if $\lim_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}$ exists then it must be $0$?
Clunie (unpublished) proved this if $a_n\geq 0$ for all $n$. This was disproved in general by Clunie and Hayman [ClHa64], who showed that the limit can take any value in $[0,1/2]$.

See also [513].

SOLVED
Does there exist, for all large $n$, a polynomial $P$ of degree $n$, with coefficients $\pm 1$, such that $\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}$ for all $\lvert z\rvert =1$, with the implied constants independent of $z$ and $n$?
Originally a conjecture of Littlewood. The answer is yes (for all $n\geq 2$), proved by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST19].

See also [230].

Additional thanks to: Mehtaab Sawhney
OPEN
Let $(S_n)_{n\geq 1}$ be a sequence of sets of complex numbers, none of which has a finite limit point. Does there exist an entire function $f(z)$ such that, for all $n\geq 1$, there exists some $k_n\geq 0$ such that $f^{(k_n)}(z) = 0\textrm{ for all }z\in S_n?$
Additional thanks to: Zachary Chase
SOLVED
Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have $\max_{\lvert z\rvert=1}\lvert P(z)\rvert \geq (1+c)\sqrt{n}?$
The lower bound of $\sqrt{n}$ is trivial from Parseval's theorem. The answer is no (contrary to Erdős' initial guess). Kahane [Ka80] constructed 'ultraflat' polynomials $P(z)=\sum a_kz^k$ with $\lvert a_k\rvert=1$ such that $P(z)=(1+o(1))\sqrt{n}$ uniformly for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$, where the $o(1)$ term $\to 0$ as $n\to \infty$.

For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with $P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})$ for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.

See also [228].

Additional thanks to: Mehtaab Sawhney
SOLVED
Let $S$ be a string of length $2^k-1$ formed from an alphabet of $k$ characters. Must $S$ contain an abelian square: two consecutive blocks $x$ and $y$ such that $y$ is a permutation of $x$?
Erdős initially conjectured that the answer is yes for all $k\geq 2$, but for $k=4$ this was disproved by de Bruijn and Erdős. At least, this is what Erdős writes, but gives no construction or reference, and a simple computer search produces no such counterexamples for $k=4$. Perhaps Erdős meant $2^k$, where indeed there is an example for $k=4$: $1213121412132124.$

Erdős then asked if there is in fact an infinite string formed from $\{1,2,3,4\}$ which contains no abelian squares? This is equivalent to [192], and such a string was constructed by Keränen [Ke92]. The existence of this infinite string gives a negative answer to the problem for all $k\geq 4$.

Containing no abelian squares is a stronger property than being squarefree (the existence of infinitely long squarefree strings over alphabets with $k\geq 3$ characters was established by Thue).

We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results.

OPEN - $500 Given$n$distinct points$A\subset\mathbb{R}^2$must there be a point$x\in A$such that $\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?$ Or even$\gg n/\sqrt{\log n}$? The pinned distance problem, a stronger form of [89]. The example of an integer grid show that$n/\sqrt{\log n}$would be best possible. It may be true that there are$\gg n$many such points, or that this is true on average. In [Er97e] Erdős offers \$500 for a solution to this problem, but it is unclear whether he intended this for proving the eixstence of a single such point or for $\gg n$ many such points.

In [Er97e] Erdős wrote that he initially 'overconjectured' and thought that the answer to this problem is the same as for the number of distinct distances between all pairs (see [89]), but this was disproved by Harborth. It could be true that the answers are the same up to an additive factor of $n^{o(1)}$.

The best known bound is $\gg n^{c-o(1)},$ due to Katz and Tardos [KaTa04], where $c=\frac{48-14e}{55-16e}=0.864137\cdots.$