39 problems

Square-prime short-interval conjecture. For all $x \geqslant 2$,

Hassani's density conjecture. For every $\epsilon>0$ there exists $n_{\epsilon}\in\mathbb N$ such that for all $n>n_{\epsilon}$ we have

Let $F(n)$ be the number of primes in the interval $(n^2,(n+1)^2)$. Hassani's lower-bound conjecture. For every $n\geq 3$, we have … The conjecture was checked computationally for…

Let $r(n)$ denote the number of ways an even integer $n=2m$ can be expressed as a sum of two primes. Let $\pi(n)$ denote the number of primes up to $n$, and let the product below e…

Let $r_k(N)$ denote the largest size of a subset of $\{1,\dots,N\}$ containing no nontrivial arithmetic progression of length $k$, and let $\pi(N)$ denote the number of primes at m…

Asymptotic expansion conjecture. There exists a constant $\kappa>0$ such that

Marques–Trojovský conjecture. The function $\Psi(x)$ satisfies: (i) $\Psi(x)=21$ for all $x\in[0.9,1.3]$; (ii) $\Psi(x)=149$ for all $x\in[0.5,0.8]$; (iii) $\Psi(x)=59{,}875$ for a…

First Hardy–Littlewood conjecture. Unless $P$ forms a complete residue class with respect to some prime, $\pi_P(x)$ is asymptotic to

Let $G(x)$ denote the number of primes in the set $\left\{\left\lfloor x/n\right\rfloor:1\le n\le x\right\}$. Suppose that $x=p_1p_2p_3$, where $p_1,p_2,p_3$ are primes satisfying…

Let $\mathcal L^+$ denote the set of positive integers associated with quadratic characters whose partial sums are positive, and let $\pi(x)$ denote the prime-counting function. De…

Quantitative twin-prime lower-bound conjecture. For each integer $a>0$ there exists an integer $N_2(a)$ such that

Shapiro-class formula for $\pi(n)$. Then $G(n)\sim\pi(n)$ for a constant $b$, where

Let $f(t)$ and $g(t)$ be strictly increasing functions tending to infinity as $t$ tends to infinity. Let $\bar D_{f,g}(t)$ count good pairs $(m,n)$ with $m,n$ prime, $m\leq f(t)$,…

Let $\bar D(t)$ count good pairs $(m,n)$ with $m,n$ both prime and $m\leq t$, $n\leq t$, where goodness means that $\Phi_m(\Phi_n(x))$ factors over the integers. Let $\pi(t)$ denot…

Let $x\geq 2$ be large, let $r>0$ be real, and set $y=\log^r x$. With $\pi(t)=\#\{p\leq t:p\text{ is prime}\}$, the oscillation conjecture. Both inequalities … and … occur infinite…

Let $\pi(x)$ denote the number of primes at most $x$, and let $x\geq2$ and $y$ be in the range intended by the source. Hardy--Littlewood's second conjecture. The conjecture asserts…

Shifted-index restricted-prime conjecture. For every fixed positive integer $k$, the sequence $(A_n^{(k)})$ satisfies the Restricted Prime Number Theorem. The conjecture is present…

Let $p_i$ be the $i$th prime and define … For a fixed nonnegative integer $d$, define the shifted sequence $A_n^{(d)}=2d+A_n$. Shifted restricted-prime conjecture. For every fixed…

Let $(A_n)$ be defined by $A_n=\sum_{i=1}^{2n}(-1)^ip_i$, and let $r_k=A_m$ be the $k$th prime occurring in this sequence. Alternating-sum prime lower-bound conjecture. For every…

Let $p_i$ be the $i$th prime, and define … For a sequence $(b_n)$, let $\pi_{(b_k)}(b_n)$ denote the number of primes among $b_1,\ldots,b_n$. The alternating-sum distribution conje…

Sondow–Nicholson–Noe conjecture. For every positive integer $m$ and every integer $n\geq N(m)$,

Let $n$ and $k$ be integers, and let $[n,kn]$ be the corresponding interval. Mitra–Paul–Sarkar's upper-bound conjecture. The number of primes in $[n,kn]$ is at most … The conjectur…

Let $n$, $a$, and $k$ be integers, where … The number of primes between $n$ and $kn$ is at least $k-1$ when $n\geq a$. Mitra–Paul–Sarkar's generalized Bertrand conjecture. For any…

A Bachet anomalous prime is a prime of good reduction for a curve of the form $y^2=x^3+D$ whose reduction has exactly $p$ points. Let $N$ be a positive real number, and let $c$ be…

Let $N(k)$ be an integer with $N(k)\geq 120$. Let $L(N(k))$ count the primes contained inside the sum of intervals created by $N(k)$, let $D(N(k))$ be the number of prime factors o…