128 problems

Let $0<\theta<1$ and $C>0$ be fixed. Let $\Lambda(n)$ denote the von Mangoldt function and $\varphi(n)$ the Euler totient function. Elliott–Halberstam conjecture. For every $A>0$ a…

Dickson's conjecture. There should be infinitely many such pairs of primes $p,q$.

Bunyakovsky's conjecture. The value $f(n)$ should be prime for infinitely many positive integers $n$.

Let $p_n$ denote the $n$th prime. Firoozbakht's conjecture. For every $n$, … The conjecture implies increasingly strong upper bounds on prime gaps, including the bounds quoted in t…

Let $p_n$ denote the $n$th prime. Andrica's conjecture. For every $n$, … This conjecture gives a strong upper bound on gaps between consecutive primes and remains open.

Let $T_n=n(n+1)/2$ denote the $n$th triangular number. Sierpiński's conjecture. Between every two consecutive triangular numbers, there lies at least one prime number. This is a pr…

Let $p_n$ denote the $n$th prime, with $n\ge 2$. Brocard's conjecture. There are at least four primes between $p_n^2$ and $p_{n+1}^2$. This is an open conjecture on the number of p…

Let $S_0(a,b)$ be the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi_{0,a,b}$ count the primes in $S_0(a,b)$ not exceeding…

Let $b$ and $l$ be coprime integers, and consider the first prime $p\equiv b\pmod l$. Granville–Pomerance conjecture. For infinitely many choices of $l$, this first prime satisfies…

Let $k$ be the length of the progression, let $d$ be its common difference, and let $A_{k,d}$ be the adjustment factor defined by the local prime-divisibility probabilities. Prime…

The Riemann hypothesis asserts that the nontrivial zeros of the Riemann zeta-function lie on the line $\Re(s)=1/2$. Generalized Riemann hypothesis. The same property holds for a mu…

Lemke Oliver–Soundararajan conjecture.

Barban–Davenport–Halberstam conjecture. Under these conditions,

Let $R_B(x)$ denote the number of $B$-regular primes up to $x$, where an odd prime is $B$-regular if it divides none of the numerators of $B_2,B_4,\ldots,B_{p-3}$. Siegel's conject…

Let a semiprime be a product $m=pq$ of two, not necessarily distinct, primes. For each $x\geq 1$, consider the semiprimes $m\leq x$ and compare those whose two prime factors satisf…

Let $\mathcal{P}$ denote the set of primes, and let $1_{\mathcal{P}}$ be its indicator function. For a tuple of distinct integers ${\mathcal H}=\{h_1,\dots,h_k\}$, let…

Let $K=\mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field, and let $C_1$ be the cone whose elements have angle between $\pi/36$ and $2\pi/36$. Let $P_M(-d)$ count primes less t…

Let $m$ be an integer. Strong Legendre conjecture. The interval … contains at least two primes. This is stronger than standard Legendre's conjecture and is presented as a consequen…

Friedlander–Goldston conjecture. The Hooley asymptotic $G(X,Q)\sim X\log Q$ should hold when $X^{1/2+\epsilon}<Q<X$, and, when $X^{1/2+\epsilon}<Q<X^{1-\epsilon}$,

Infinitude conjecture. Infinitely many primes satisfy this congruence.

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…

Smooth-number distribution conjecture.

For a positive integer $n$, let $H(1;n)=\sum_{k=1}^n1/k$, and for a prime $p$ let $J(1|p)$ denote the set of indices whose reduced numerator is divisible by $p$. Eswarathasan–Levin…

Let a positive integer $n$ be 2-good if $n-1$ has a prime factor $p$ satisfying … For an integer interval, let the numerator below count the 2-good primes in it and the denominator…

Let $g\notin\{-1,0,1\}$ be rational, and let $N_g(a,d)$ denote the set of primes $p$ for which the residual order $\operatorname{ord}_g(p)$ is congruent to $a$ modulo $d$. For the…