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,…

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 $p_n$ 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…

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

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. 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 $n$ be an integer with $n>1$. Oppermann's conjecture. There is at least one prime number between $n(n-1)$ and $n^2$, and at least another prime number between $n^2$ and…

Let $p_n$ denote the $n$-th prime. Cramér's conjecture. There exist absolute constants $n_0,c>0$ such that if $n>n_0$, then … This conjectural bound on maximal gaps between consecu…

Let $p$ and $q$ be consecutive prime numbers other than the exceptional pair $2$ and $3$. Brocard's conjecture. There are at least four primes strictly between $p^2$ and $q^2$. Thi…

Odlyzko–Rubinstein–Wolf conjecture. The jumping champions tend to infinity. Furthermore, any fixed prime $p$ divides all sufficiently large jumping champions.

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 $\mathbb{P}$ 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…

Let $\mathcal{D}_k^1=L_{\infty}(p_{k+1})-L_{\infty}(p_k)$ be the prime gaps along the number trail, and let $\mathbb{N}$ denote the natural numbers. Modified Polignac's conjecture.…

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 $p_n$ 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…

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