Open-problem statements, with their sources attached.

In particular, is it true that $\epsilon_n = o(1)$?

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 the congruences ≡ a_p (mod p)?

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.

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

Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and

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

Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j), one can always take p > i in the prime divisor statement.

Is there a covering system all of whose moduli are odd (and greater than 1)?

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$ and let $P(n)$ be the largest prime dividing $n$. (a)* Characterise those composite $n$ such that $f(n) = n/P(n)$.

Erdős–Szekeres [ErSz78] note that $f(n) = n/P(n)$ when $n$ is a product of two primes (erdos_700.variants.prime_mul), with $n = 30$ a further example. The characterisation itself is open; we state it as the (unknown) predicate that is equivalent to being such an n.

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$. (b)* Are there infinitely many composite $n$ such that $f(n) > n^{1/2}$?

Erdős–Szekeres [ErSz78] could not prove this. (Since $f(n) \ge p(n)$, the least prime factor of $n$, there are infinitely many $n$ , those of the form $p^2$ , with $f(n) \ge n^{1/2}$; the question asks for the strict inequality.) Here $f(n) > n^{1/2}$ is written as (f n) ^ 2 > n.

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$. (c)* Is it true that, for every composite $n$, $f(n) \ll_A n/(\log n)^A$ for every $A > 0$?

Erdős–Szekeres [ErSz78] prove the weaker bound $f(n) \le (1 + o(1)) n/\log n$ (the case $A = 1$). Here $f(n) \ll_A n/(\log n)^A$ is spelled out as: for every A > 0 there is a constant C (depending on A) with f(n) ≤ C · n/(log n)^A for every composite n.

As $n\to \infty$ ranges over integers $\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}$?

A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75].

By $n\in (p/2,p)\pmod{p}$ we mean $n\equiv r\pmod{p}$ for some integer $r$ with $p/2<r<p$.

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

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

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 $n$?

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.