Browse Problems

Greaves, Harman, and Huxley [GHH97] showed that this is true for $0 \leq \alpha\leq 11/3$. [GHH97] Greaves, G. R. H. and Harman, G. and Huxley, M. N., Sieve Methods, Exponential Sums, and …

Hooley [Ho73] extended this to all $0 \leq \alpha\leq 3$. [Ho73] Hooley, Christopher, On the intervals between consecutive terms of sequences. Proc. Symp. Pure Math, vol. 24, pp. 129-140. 1973.

Erdős [Er51] proved this for all $0\leq \alpha\leq 2$. [Er51] Erdös, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.

Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ be the set of integers which are representable in exactly one way as the sum of two elements from $A$. Is it true that …

Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?

Must `lim f n = ∞`?

Must `f n ≫ n ^ 2`?

Must `lim f n = ∞`?

Is it true that for every $k \geq 1$ we have $$ F(N + k) \leq F(N) + 1 $$ for all sufficiently large $N$?

Let `A` be an infinite `B₂[2]` set. Must `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0`?

As a corollary of `erdos_158.isSidon'`, we can prove that `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0` for any infinite Sidon set `A`.

Let `A` be an infinite Sidon set. Then `liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) * (log N) ^ (1 / 2) < ∞`. This is …

Estimate $h(n)$ by finding a better lower bound.

Estimate $h(n)$ by finding a better upper bound.

The observation of Zachary Hunter in [that question](https://mathoverflow.net/q/410808) coupled with the bounds of Kelley-Meka [KeMe23](https://arxiv.org/abs/2302.05537) imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$.

On [Mathoverflow](https://mathoverflow.net/a/410815) user [leechlattice](https://mathoverflow.net/users/125498/leechlattice) shows that $h(n) \ll n^{\frac 2 3}$.

What is the limit $F(N)/N$ as $N \to \infty$?

Is the limit $F(N)/N$ as $N \to \infty$ irrational?

The limit $F(N)/N$ as $N \to \infty$ exists. (proved by Graham, Spencer, and Witsenhausen)

**Erdős Problem 17.** Are there infinitely many cluster primes?

The problem is to determine the limit of the sequence $\frac{F(N)}{\sqrt{N}}$ as $N \to \infty$.

The existence of the limit has been proved by Erdős and Gál [ErGa48]. The lower bound has been proven by Leech [Le56], who refined an argument of Rédei and Rényi. The upper …

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?

In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound $$\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$$ for every real $A > 0$. [BES99] Blecksmith, Richard and Erd\H os, Paul and Selfridge, J. L., …

In 2003, Elsholtz [El03] refined the upper bound to $$\pi^{\mathcal{C}}(x) \ll x\,\exp\!\bigl(-c(\log\log x)^2\bigr)$$ for every real $0 < c < 1/8$. [El03] Elsholtz, Christian, On cluster primes. Acta Arith. (2003), 281--284.

What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?

Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15): How small can $M$ be made? The only estimate currently known is that $M \le 10000000$ …

Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15): It has been shown that there is a large $M$ so that it is possible …

If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area? Graham, "On Partitions of 𝔼ⁿ", Journal of Combinatorial Theory, Series …

Seems to be open, as of January 2025.

Graham claims this is "easy to see".

Let $k\geq 3$. Must any ordering of $\mathbb{R}$ contain a monotone $k$-term arithmetic progression, that is, some $x_1 <\cdots < x_k$ which forms an increasing or decreasing $k$-term arithmetic progression? The answer …

What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1 < \cdots < x_k$?

Adenwalla [Ad22] proved that k ≤ 4.

Geneson [Ge19] proved that k ≤ 5.

Must every permutation of $\mathbb{N}$, contain a monotone 4-term arithmetic progression?

Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?

If $A \subseteq \mathbb{N}$ is a Sidon set then must the complement of $A$ contain an infinite arithmetic progression? Answer "yes" according to remark on page 23 of: - Erdös and Graham, …

In fact one such sequence is $n! + n$. This was found by AlphaProof. It also found $(n + 1)! + n$.

Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?

Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?

It follows from the prime number theorem that such a progression has length $\leq(1+o(1))\log N$.

Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \cdot 3^\ell \cdot m + 1$ are prime, for any $k, \ell \ge 0$?

Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'? That is, for all $d\mid n$ with $d>1$ there is …

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that for any $\epsilon > 0$ and large $n$, $s_{n+1} - s_n \ll_\epsilon s_n^\epsilon$?

Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers. Is it true that $s_{n + 1} - s_n \le (1 + o(1)) \cdot (\pi^2 / 6) \cdot \log (s_n) …

In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful'. [Er79] Erdős, Paul, __Some unconventional problems in number theory__. Math. Mag. (1979), 67-70.

Is there a dense subset of ℝ^2 such that all pairwise distances are rational?

Let $n \geq 4$. Are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers?

The best construction to date, due to Kreisel and Kurz, has $n = 7$.