239 problems

Let the preceding Ova-Ova-Prime-Ova construction be given, and let $N$ denote the number of primes it produces. The Ova-Ova-Prime-Ova counting conjecture. Although the preceding th…

The binary Goldbach conjecture asserts that every even integer greater than $2$ is the sum of two primes:…

For every prime $p\ge 3$, there exist distinct primes $p_1$ and $p_2$ such that $p=2p_1-p_2$; equivalently,…

$\#\{p\in\mathbb{N}: p\text{ and }p+2\text{ are prime}\}=\infty.$

Let $A\subseteq\mathbb{N}$ be admissible if there is no prime $p$ such that $A$ contains at least one element in every residue class modulo $p$. Erdős and Graham's conjecture. Ther…

Are there infinitely many primes $p$ such that $p-k!$ is composite for each $k$ such that $1\leq k!<p$?

For $x\in\mathbb R$, define … where $p_n$ is the $n$th prime, $\pi'$ is the natural-number prime-counting function, and the supremum is taken in $\mathbb N$ (equivalently, this is…

For $0\leq k\leq n$ write $\binom{n}{k} = uv$ where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be the smallest $k$ su…

Let $p_1 < p_2 < p_3 < \cdots$ denote the increasing sequence of all prime numbers, so that $p_k$ and $p_{k+1}$ are consecutive primes for every $k \ge 1$. For an even positive int…

Let $\mathbb{P}=\{2,3,5,7,11,\dots\}$ denote the set of prime numbers, and let $p_{1}<p_{2}<p_{3}<\cdots$ be the elements of $\mathbb{P}$ listed in increasing order. Call a pair of…

Let $\mathbb{Z}[i]=\{a+bi : a,b\in\mathbb{Z}\}$ be the ring of Gaussian integers, with norm $N(a+bi)=a^{2}+b^{2}$ and Euclidean absolute value $|a+bi|=\sqrt{a^{2}+b^{2}}$. Call…

Let $k \ge 1$ and let $a_1, \dots, a_k$ and $b_1, \dots, b_k$ be integers with $b_i \ge 1$ for every $i$, and consider the $k$ linear forms … Suppose that no prime divides all the…

Let $\epsilon,\eta>0$. Does there exist $k=k_0(\epsilon,\eta)$ such that, whenever $p_1<\cdots<p_k<n^{1-\epsilon}$ are primes and $A_i$ consists of $(p_i-1)/2$ residue classes modu…

If $f(n)$ denotes $\min_i (p_{n+i} + p_{n-i})$, is it true that …

Perhaps the following rather silly conjecture could be added. Is it true that the set of odd integers not of the form $2^k + p$ is the not necessarily disjoint union of an infinite…

For $n\in\mathbb{N}$, let … A prime $p$ satisfying $w_p\equiv 1\pmod{p^3}$ is considered in the context of Wolstenholme's theorem. Jones' conjecture. … This asserts that the known…

For $n\in\mathbb{N}$, let $P(n)$ mean that for every $k\in\mathbb{N}$, if $k^2<n$ and $\gcd(n,k)=1$, then $n-k^2$ is prime. Are there infinitely many natural numbers $n$ satisfying…

Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$,…

Let $A\subseteq\mathbb N$ satisfy $|A\cap\{1,\ldots,N\}|\gg\log N$ for all sufficiently large $N$, and let $f(n)$ count representations $n=p+a$ with $p$ prime and $a\in A$. Must…

Consider the graph with vertex set consisting of pairs $(x,y)\in\mathbb N^2$ such that … and at least one of $x$ or $y$ is nonprime. Two vertices $p=(p_1,p_2)$ and $q=(q_1,q_2)$ ar…

For $\epsilon,\eta>0$, let $P\bigl(n(n+1)\cdots(n+k)\bigr)$ denote the greatest prime divisor of the product. Is it true that there exists $k\in\mathbb N$ such that the lower asymp…

Is there an absolute constant $C$ such that for every sufficiently large $X$ one can choose primes $p_i<X$ with $\sum_i1/p_i<C$ and residue classes $a_i\pmod{p_i}$ whose union cont…

For each $k≥3$, does every finite colouring of the positive integers contain a monochromatic $k$-term arithmetic progression all of whose terms are prime? Does it contain a monochr…

For $n\in\mathbb N$, let $P(n)$ mean … Is the set of natural numbers $n$ satisfying $P(n)$ infinite?

Let $1 \leq u_0<u_1<u_2<\cdots$ be the increasing sequence of positive integers having at most two prime factors, counted with multiplicity. Is …