45 problems

Let $h_1,h_2,\ldots,h_k$ be distinct positive even integers such that the numbers in the sequence $\mathbb{P}_k=(p,p+h_1,\ldots,p+h_k)$ do not form a complete residue class modulo…

Let $\mathcal{L} = \{L_1,\ldots,L_K\}$ be a set of distinct linear forms $L_k(n) = a_kn+b_k$, where the coefficients $a_k,b_k$ are positive integers. For a prime $p$, let…

Dickson–Hardy–Littlewood conjecture. If $H\subset\mathbb{N}$ is finite and admissible, then there exist infinitely many $n\in\mathbb{N}$ such that

Let $\mathcal A=\{a_1,\ldots,a_s\}$ be an admissible set of distinct natural numbers, meaning that the associated linear forms satisfy the usual local non-obstruction condition. Ch…

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 $L$ be a sublattice of $\mathbb{Z}^t$, let $N$ be large, let $K\subseteq[-N,N]^t$ be convex with volume $\gg N^t$, and let $a\in\mathbb{Z}^t$ have size $O(N)$. Assume that all…

Let $L$ be an affine sublattice of $\mathbb{Z}^t$. For each prime $p$, let $\alpha_p(L)$ be the local factor defined from the image $L_p$ of $L$ in $\mathbb{Z}_p^t$; a local obstru…

Let $N,d,t,L$ be positive integers, let $\Psi=(\psi_1,\ldots,\psi_t)$ be a system of affine-linear forms with $\|\Psi\|_N\leqslant L$, and let $K\subset[-N,N]^d$ be a convex body.…

Let $\Lambda(n)$ be the von Mangoldt function, and for distinct integers $d_1,\ldots,d_k$ with $1\leq d_i\leq H$, let $\mathcal{D}=\{d_1,\ldots,d_k\}$. Define $E_k(x;\mathcal{D})$…

Let $A>0$ be an integer. Suppose that $(M_1x+a_1)(M_2x+a_2)$ has no fixed prime divisors as $x$ ranges over the integers, with $M_1,M_2>0$ and $0\leq a_i<M_i$ for $i=1,2$. Strong H…

McCranie's conjecture. Every composite integer $n$ satisfying

Let $W\geq 1$ be fixed, and let $b_1,b_2$ be fixed distinct integers coprime to $W$. For each prime $p$, let $\nu_p(W\cdot+b_1,W\cdot+b_2)$ be the number of distinct residue classe…

Specific case of prime tuplets conjecture. The number of integers $q$ with $x\leqslant q<2x$ for which every $qr_j+1$ is prime satisfies

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

Odd-moment conjecture. As $h$ tends to infinity,

Let $x\to\fty$. Let $a_0,a_1,\ldots,a_{k-1}$ be a fixed admissible $k$-tuple, and let $b_0<b_1<\cdots<b_{l-1}$ be a fixed subset of small integers. Hardy–Littlewood–Chowla conjectu…

Prime-tuple hypothesis. If no congruence obstruction exists, then there are infinitely many values of $x$ such that every $a_i x+b_i$ is prime.

A finite set of integers $\{a_1,a_2,\ldots,a_k\}$ is admissible if, for every prime $q$, its residue classes do not cover all residue classes modulo $q$. Quantitative extension of…

For $k\geq0$, define … where $\Lambda$ is the von Mangoldt function and $\mathfrak{S}(k)$ is the singular series. The weakened prime-pair upper-bound conjecture. Given a fixed cons…

Cunningham chain conjecture. The number of Cunningham chains of length $k$ beginning with a prime $p\leq N$ is approximately

Skewes-number conjecture. Every admissible prime $k$-tuple has a Skewes number. The conjecture is based on the computational results in the paper; no proof or disproof is supplied.

Wolf's conjecture. The number of sign changes of $d_2(n)$ for $n\in(1,T)$ is $\frac{\sqrt T}{\log T}$. This numerical prediction concerns fluctuations around the Hardy–Littlewood e…

Let $(a_1,a_2,\ldots,a_k)$ be a monotonically increasing sequence of positive even integers. Let $p$ be prime, and let $P=(p,p+a_1,p+a_2,\ldots,p+a_k)$ be an admissible prime $k$-t…

Fix $R\in\mathbb{N}$ and $C>0$. Let $a_i,b_i\in\mathbb{Z}$ satisfy $(a_i,b_i)=1$ and $a_i>0$ for $1\leq i\leq R$. Define the Hardy–Littlewood singular series … where $\rho(p)$ is t…

Let $\mathbb{S}$ be the set of integers expressible as a sum of two squares. For $k\geq1$ and a set $\boldsymbol{h}=\{h_1,\ldots,h_k\}\subseteq\mathbb{Z}$ with…