Open-problem statements, with their sources attached.

Conjecture 1.* Are there infinitely many practical numbers $m$ such that $h(m) < (\log \log m)^{O(1)}$?

More precisely: does there exist a constant $C > 0$ such that for infinitely many practical numbers $m$, we have $h(m) < (\log \log m)^C$?

Conjecture 2.* Is it true that $h(n!) < n^{o(1)}$? That is, for all $\varepsilon > 0$, is $h(n!) < n^\varepsilon$ for sufficiently large $n$?

Conjecture 3.* Or perhaps even $h(n!) < (\log n)^{O(1)}$?

Erdős offered $250 for a proof or disproof.

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 $s_{n + 1} - s_n \le (1 + o(1)) \cdot (\pi^2 / 6) \cdot \log (s_n) / \log (\log (s_n))$?

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.

The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$.

The set of indices $n$ for which a prime gap is preceeded by a larger or equal prime gap has a natural density of $\frac 1 2$.

There are infintely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$. This is equivalent to the existence of infinitely many arithmetic progressions of length $3$, see erdos_141.variants.infinite_three.

A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$.

Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?

Let c₁, c₂ > 0. Is it true that for any sufficiently large x, there exists more than c₁ * log x many consecutive primes ≤ x such that the difference between any two is > c₂?

For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$.

Schinzel conjectured (see [Si56]) the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x < y < z$ such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$

Let $C > 1$. Does the set of integers of the form $p + \lfloor C^k \rfloor$, for some prime $p$ and $k\geq 0$, have density $>0$?

Let $n_1 < n_2 < \cdots$ be a sequence of integers such that

Is

transcendental?

Is $\sum_{n} \frac{\phi(n)}{2^n}$ irrational? Here $\phi$ is the Euler totient function.

Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n < n_i$ or $n \not\equiv a_i \pmod{n_i}$. Must the logarithmic density of $A$ exist?

Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational? Here $p_n$ is the $n$-th prime ($p_1=2, p_2=3, \dots$).

Erdős Problem 252: irrationality of the sum for a given $k$.