76 problems

Let $U(n)$ be the maximum number of unordered pairs of points at Euclidean distance $1$ among any set of $n$ points in the plane, that is,…

Polylogarithmic prime-gap conjecture. There exists an absolute constant $k>1$ such that

Let $\mathcal{P}_{\mathrm{good}}=\{g_1,g_2,\ldots\}$ be the subset of good primes, enumerated in increasing order, where good primes are the primes outside the paper's exceptional…

Let $\mathcal{P}=\{p_1,p_2,\ldots\}$ be the set of prime numbers, enumerated in increasing order. Cramér–Granville conjecture. For some $\kappa>1$, … Cramér proposed the correspond…

Folklore equality conjecture. The inequality is in fact an equality:

Let $\pi(t)$ denote the number of primes at most $t$. For a real parameter $\lambda=(\log x)^{o(1)}$ and an integer $k\ll (\log_2 x)^2$, count integers $n\leqslant x$ according to…

Let $p_k$ be a prime, let $\mathcal{G}(p_k^{\#})$ be the corresponding Eratosthenes-sieve cycle, and let $\Delta H(p_k)=[p_k^2,p_{k+1}^2]$ be its interval of survival. The populati…

Define $q_1=2$, and for each positive integer $n$, let $q_{n+1}$ be the least prime exceeding $q_n^2$. Legendre's recursive prime-gap assertion. Then … for all $n$. This assertion…

For each integer $n\geq 1$, consider the interval $(n^3,(n+1)^3)$. Prime-between-consecutive-cubes conjecture. For all $n\geq 1$, there exists a prime between $n^3$ and $(n+1)^3$.…

Let $p_n$ be the $n$-th prime, with $p_0=1$, and let … be the first $n$ prime gap sequence. A prime gap graph on $n\geq 2$ vertices is a simple graph whose vertex degrees are exact…

Let $p_n$ denote the $n$th prime, and let $\varepsilon>0$. Poisson Tail Conjecture. For $1\leq H\leq(\log X)^{2-\varepsilon}$, the two gap-counting quantities satisfy … … For…

Let $\chi$ be the chaotic map used in the EMCHS model, let $y_0\in[0,1]$, and let $\rho(y)$ denote the proposed invariant density. For a suitable class of functions $g$, bounded-ch…

Let $p_n$ denote the $n$th prime, and let $\delta$ and $\epsilon$ be parameters in the ranges considered by the Enhanced Multidimensional Chaotic Heuristic Sieve (EMCHS), including…

Let $q_1,q_2,\ldots$ be the consecutive prime powers. Andrica's conjecture for prime powers. The inequality … holds for every $n$. This is a denser-sequence analogue of Andrica's c…

Let $\mathbb{P}$ denote the set of primes, and let $\mathbf{POL}$ be the set of even numbers that occur infinitely many times as differences of two consecutive primes. Polignac's c…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Kronecker's conjecture. Every even number can be expressed in infinitely many ways as the…

Let $\mathbb{P}$ denote the set of primes. An even number is an element of $2\mathbb{N}$. Maillet's conjecture. Every even number is the difference of two primes. This is a foundat…

Sierpiński's conjecture. Every row of this matrix contains at least one prime number.

Legendre's conjecture. There always exist at least two prime numbers between $n^2$ and $(n+1)^2$.

Let $p_n$ denote the $n$th prime. Twin-square-interval conjecture. For every integer $n\geq 3$, there exists at least one pair of twin primes lying in the interval … This is a stro…

Let $p_n$ denote the $n$-th prime number, with $p_0=1$. A prime gap graph on $n$ vertices is a simple graph whose vertex degrees are $p_1-p_0,\dotsc,p_n-p_{n-1}$; two edges are ind…

Let $S$ be the set of odd primes, and for $a,b\in S$ let $a\bullet b$ denote the smallest prime strictly greater than $|a-b|$. Loop equation conjecture. For every $x\in S$, the equ…

Let $p_n$ be the $n$th prime, let $\mathcal P$ denote the set of partitions, and let $\widehat{N}$ be the supernorm map. A partition has no $1$'s when none of its parts equals $1$,…

Finite Gilbreath conjecture. For every such prime originator,

Let a series be formed from reciprocals associated with twin primes and with prime pairs separated by $2k$. The reciprocal-series conjecture. The series of reciprocals of twin prim…