Open-problem statements, with their sources attached.

If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?

Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?

It is conjectured that $f(k) \ll (\log k)^O(1)$.

Main conjecture:

$\log k(n) \le (\log n)^{(1/2 + o(1))}$

Does the set {n | u n < u (n+1)} have positive natural density?

Erdős asked whether there are infinitely many solutions to uₙ < uₙ₊₁ < uₙ₊₂.

Erdős asked whether there are infinitely many solutions to uₙ > uₙ₊₁ > uₙ₊₂.

Let p(a, d) be the least prime congruent to a (mod d). Does there exist a constant c > 0 such that for all large d, p(a, d) > (1 + c) * φ(d) * log d for ≫ φ(d) many values of a?

Erdős problem 972.* Let $\alpha > 1$ be irrational. Are there infinitely many primes $p$ such that $\lfloor p\alpha \rfloor$ is also prime?

Does there exist a constant $C>1$ such that, for every $n\geq 2$, there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with $\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}$?

This is Problem 7.3 in [Ha74], where it is attributed to Erdős.

For an irreducible polynomial $f \in \mathbb{Z}[x]$ with $f(n) \ge 1$ for sufficiently large $n$, does there exists a constant $c = c(f) > 0$ such that $\sum_{n \le x} \tau(f(n)) \approx c \cdot x \log x$?

Note that it is unclear whether the polynomial should have integer coefficients or merely be integer-valued. We assume the former.

If $k>3$ (and $k \neq 2^l$), and for all primes $p$ there exists $n$ such that $p^{k-2}\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?

Does n ^ 4 + 2 represent infinitely many squarefree numbers?

Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$, where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n) = \infty$?

Is it true that, for every prime $p$, there is a prime $q \leq p$ which is a primitive root modulo $p$?

Let $A \subset \mathbf{Z}$ be a set of $n$ integers. Is there a set $S \subset A$ of size $(\log n)^{100}$ such that the restricted sumset$S \hat{+} S$ is disjoint from $A$?

Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$. Is there a solution to $xy = z$ with $x, y, z \in A$?

Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$. Does the remaining set have size at most $\frac{1}{10} N$?

We interpret "half the residue classes" as $\lfloor p_i / 2 \rfloor$.

We conjecture that the best-known lower bound can be improved.

We conjecture that the best-known upper bound can be improved.