Browse Problems

Is it true that, for all sufficiently large $n$, there exists some $k$ such that \[ p(n+k)>k^2+1, \] where $p(m)$ denotes the least prime factor of $m$?

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?

**Erdős problem 681.** Is it true that for all large $n$ there exists $k$ such that $n + k$ is composite and $p(n+k) > k^2$, where $p(m)$ is the least prime factor …

Can every integer $N≥2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?

Let `n` be sufficiently large. Is there some choice of congruence class `a_p` for all primes `2 ≤ p ≤ n` such that every integer in `[1,n]` satisfies at least two of …

Let $f_\max(n)$ be the largest $m$ such that $\phi(m) = n$, and $f_\min(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function. Investigate $$ \max_{n\leq x}\frac{f_\max(n)}{f_\min(n)}. $$

Carmichael has asked whether there is an integer $n$ for which $\phi(m) = n$ has exactly one solution, that is $\frac{f_\max(n)}{f_\min(n)} = 1$.

Erdős has proved that if there exists an integer $n$ for which $\phi(m) = n$ has exactly one solution, then there must be infinitely many such $n$.

Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$. Is it true that $$ \lim_{k \to \infty} q_k^{1/k} = \infty? $$

Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $$ q(k) \leq \exp(k (\log k)^{1 + o(1)})? $$

Let $\beta = \frac{1}{\log 2}$. Then $lim_{m\rightarrow\infty} \delta (m, \alpha) = 0$ if $\alpha < \beta$. This is proved in [Ha92].

$\delta < \frac{m ^ \alpha + 1}{m}`. This shows that $lim_{m\rightarrow\infty} \delta (m, \alpha) = 0$ for $\alpha < 1$. #TODO: prove this theorem.

$lim_{m\rightarrow\infty} \delta (m, \alpha) = 1$ if $\beta < \alpha$. This is proved in [Ha92].

**Erdős Problem 699.** Is it true that for every $1 \le i < j \le n / 2$ there exists a prime $p \ge i$ with $p \mid \gcd\big(\binom{n}{i}, \binom{n}{j}\big)$?

Tao observed that `erdos_69` is a special case of `erdos_257`, since $$ \sum_{n\geq 2}\frac{\omega(n)}{2^n} = \sum_p \frac{1}{2^p - 1}. $$

For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \dots > d_{n+m}$, where $d$ denotes the prime gap function. Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an …

For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \dots < d_{n+m}$, where $d$ denotes the prime gap function. Proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with …

Let $G$ be a finite unit distance graph in $\mamthbb{R}^2$. Is there some $k$ such that if $G$ has girth $≥ k$, then $\chi(G) ≤ 3$? The general case was solved by …

**Erdős Problem 707**: It is false that any finite Sidon set can be embedded in a perfect different set modulo some $n$. As described in [arxiv/2510.19804], a counterexample is provided in [Ha47], …

This conjecture was actually first disproved by Hall in 1947 [Ha47], long before Erdős asked this question. A counterexample for any modulus from from [Ha47] in the paragraph following Theorem 4.3, where …

Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, showing that $\{1, 2, 4, 8, 13\}$ cannot be extended to any perfect difference set.

Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, proving that $\{1,2,4,8\}$ cannot be extended to a perfect difference set modulo $p^2+p+1$ for any prime $p$.

It is false that any finite Sidon set can be embedded in a perfect difference set modulo `p^2 + p + 1` for some prime `p`. As described in [arxiv/2510.19804], a counterexample …

It is false that any finite Sidon set can be embedded in a perfect difference set modulo `p^2 + p + 1` for some prime power `p`. As described in [arxiv/2510.19804], a …

If there is a finite projective plane of order $n$ then must $n$ be a prime power?

It is open whether there exists a projective plane of order 12.

This conjecture has been proved for $n \leq 11$.

These always exist if $n$ is a prime power.

Let $k ≥ 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many $n$?

Balakran proved this holds for $k = 1$. Let $k = 1$. Does $((n+k)!)^2∣(2n)!$ for infinitely many $n$?

Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$

It is open even for $k = 2$. Let $k = 2$. Does $((n+k)!)^2∣(2n)!$ hold for infinitely many n?

Let $\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that $$a, b > \varepsilon n\quad a!\, b! \mid n!\, (a + b - n)!, …

Are there infinitely many pairs of integers $n < m$ such that $\binom{2n}{n}$ and $\binom{2m}{m}$ have the same set of prime divisors?

There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$. (Found by AlphaProof, although it was implicit already in [A129515])

Let $f(n)\to \infty$ possibly very slowly. Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $f(n)$ edges?

Is there a basis $A$ of order $2$ such that if $A=A_1\sqcup A_2$ then $A_1+A_1$ and $A_2+A_2$ cannot both have bounded gaps?

Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive density. Can one always decompose $A=A_1\sqcup A_2$ such that $A_1+A_1$ and $A_2+A_2$ both have positive density?

Let $\epsilon>0$. Does there exist $A\subseteq \mathbb{N}$ such that the lower density of $A+A$ is at least $1-\epsilon$ and yet $1_A\ast 1_A(n) \ll_\epsilon 1$ for all $n$?

Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $\sqrt{n}$ edges?

What is the supremum of the set of admissible numbers?

The supremum is strictly larger than `1 / 2`, which is proved in [GyLe95].

In [GyLe95], the authors also prove that the supremum is smaller than `3 / 5`.

For every prime `p`, does the density of integers with `h n = p` exist?

Does `liminf h n = ∞`?

Is it true that if `p` is the greatest prime such that `p - 1 ∣ n` and `p > n ^ ε`, then `h n = p`?

It is probably true that `h n = 3` for infinitely many `n`.

Let $\epsilon > 0$. Is there some set $A\subset\mathbb{N}$ of density $> 1 - \epsilon$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = …

Is there some set $A\subset\{1, ..., N\}$ of size $\geq (1 - o(1))N$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?