179 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…

Let $(p_k)_{k\ge 1}$ be the sequence of primes in increasing order. For each integer $n\ge 2$, define $f(n)=\min_{1\le i<n}(p_{n+i}+p_{n-i})$. Is it true that…

Let $r_{p_0}$ be a ray with $p_0\in\overline{C}$. Let $l$ denote the number of correct decimal digits available in the floating-point arithmetic, and let $n^*(l)$ be a finite index…

Prime-denominator approximation conjecture. For every real $\xi$, there exists a constant $c(\xi)$ such that the inequality

Let $s,t$ be positive integers with $t\geqslant s+2$. An $s\times t$ integer matrix $\mathbf{A}$ is non-degenerate if it has rank $s$ and every non-zero vector in its row span over…

For a real number $\theta>0$ and sufficiently large $x$, the interval $[x,x+x^\theta]$ contains a prime number. The short-interval prime conjecture. Given $\theta>0$, for all suffi…

Let $(a(n))_{n\geq 1}$ be the EKG sequence defined by $a(1)=1$, $a(2)=2$, and, for $n\geq 3$, letting $a(n)$ be the smallest natural number not already in the sequence such that…

Wisde-prime infinitude conjecture. The set $\mathcal{W}$ is infinite. The paper gives computational evidence for many such primes and notes that this is equivalent to the infinitud…

Hasler's structural infinitude conjecture. The equation $\binom{3n}{n} \equiv 3^n \pmod{n}$ has infinitely many solutions of the form $n=3^k\cdot p$, where $k$ is a positive intege…

Bateman–Selfridge–Wagstaff conjecture. If two of these three statements are true, then the third is also true.

Let $\chi$ be a Dirichlet character modulo $q greater than or equal to 1$, and let ${\mathcal N}(\frac{1}{2},\infty,\chi)$ denote the number of nontrivial zeros of the associated Diric…

Let $k\in\boldsymbol{N}_{\geq 1}$ and let $bar f=(f_i:i<k)$, where $f_i(x)=a_i x+b_i$ with $a_i,b_i$ non-negative integers and $a_igeq 1$. Assume that no integer $ngeq 1$ divides al…

Let $P_{\text{ess}}$ be the essential fractal prime set, with information measure $\iota(P_{\text{ess}})=-\zeta(1/2)$, and let $Z_F$ be the fractal zero set constructed from the po…

For a base $b\geq 2$ and a positive integer $\ell$, let $\mathrm{Trun}^L_b$ denote the number of left truncations and let $\mathrm{Var}_{b^\ell}(\mathrm{Trun}^L_b)$ be its variance…

Let $\mathbb{P}(x)$ be the set of prime numbers not exceeding $x$, and let $M_{\mathbb{P}(n)}(n)$ denote the corresponding prime-restricted maximum packing number. Prime-restricted…

Let $n>1$ be a positive integer such that … for every integer $a$. The converse to Fermat's little theorem. Then $n$ must be a prime number. This statement is the converse formulat…

Let $p_n$ denote a prime number, and for each integer $k\geq 1$ define … where $p_{n+i}$ are consecutive primes. Infinite admissible lengths conjecture. For every prime number…

For an even integer $2m$, let $P(2m)$ be the set of primes $p$ such that $p-1\mid 2m$, so that … where $D_{2m}$ is the denominator of the Bernoulli number $B_{2m}$. Let…

Let $T(n)$ be the set of Toda primes of $n$, where a Toda prime is an odd prime $p$ satisfying … Write $t(n)=|T(n)|$. Toda-prime lower-bound conjecture. Let $n$ be an odd, square-f…

Under the restricted factorization described in the source, assign a fractional order $\operatorname{ord}(p)$ or $\operatorname{ord}(l)$ recursively to primes $p$ and lucky numbers…

Let $q$ be a prime satisfying $q>3$ and $q\equiv 3\pmod 4$. Simultaneous-primes conjecture. There are infinitely many numbers $3q$ such that both $9q^2-2$ and $9q^2+2$ are prime. T…

Let $a$ be the fixed shift appearing in the statement, let $P^{+}(m)$ denote the largest prime factor of $m$, and let $\boldsymbol{\vartheta}$ and $\boldsymbol{\Theta}$ be real num…

Let $n>0$ be an integer and let $x\in\{0,1\}^n$ be a vector of bits of length $n$. For each $1\leq k\leq n$, let $q_k$ denote the $k$th odd prime. Prime smoothness and prescribed v…

Write $\mathbb{P}$ for the set of primes. Prime-minus-one sum-product conjecture. For every $a\in\mathbb{N}$ there exist $x,y\in\mathbb{N}$ with $x>y>a$ such that … This asks for s…

Prime-rotation conjecture. The prime rotation sequence $(p_n\alpha)_{n \in \mathbb{N}} \mod 1$ has Poissonian correlations of all orders and, consequently, has Poissonian gap distr…