89 problems

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…

Let $j$ and $k$ be positive integers with $j\leq k$. An increasing sequence of $k$ integers is admissible if it does not occupy every residue class modulo any prime $p$; a translat…

Uniform Hardy–Littlewood prime $k$-tuple conjecture. There exists an absolute constant $0<\delta<1/2$ such that

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…

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…

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 $n$ be a positive integer, and let $\pi_{f_2}(x)$ denote the counting function for integers $n$ such that $n^2+1$ and $n^2+3$ are both prime. For each prime $p\geq 5$, let…

Strong Hardy–Littlewood prime tuple conjecture. For every fixed $k$, the prime-tuple asymptotic formula holds with an error term at most $O_k(x^{1/2+\varepsilon})$, uniformly for…

Let $P_1$ be the first principal-ray set, and let $T_1,T_2,T_3,T_4$ be the specified sets of prime pairs and related prime configurations. Define … For each $i$, let $p_{i1}$ and…

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 $E/\mathbf{Q}$ be an elliptic curve without complex multiplication, and let $j(X)$ denote the number of elliptic twins associated with $E$ up to $X$. Asymptotic conjecture. The…

Let $\mathcal{H}=\{h_1,\ldots,h_k\}$ be a set of distinct non-negative integers, and let $\nu_p(\mathcal{H})$ be the number of distinct residue classes modulo $p$ occupied by its e…

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 $E_k(x;\mathcal{D})$…

A pair of primes $q,p$ is a nearly doubled prime pair when $p=2q-1$; equivalently, $p$ is a prime such that $(p+1)/2$ is also prime. Let $N_{\mathrm{nd}}(N)$ denote the number of s…

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

Let $\mathbb{P}$ be the set of primes. Weak uniform Hardy–Littlewood conjecture. There exists a fixed $\delta>0$ such that, uniformly for every even integer $h\leq x^\delta$ and al…

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…

A Cunningham chain of length $l$ is an $l$-tuple $(p_1,\cdots,p_l)$ of distinct odd primes satisfying $p_1>\cdots>p_l$ and … for each $1\leq i<l$. Cunningham-chain infinitude conje…

Let $n$ range over the positive integers. Twin prime conjecture. There are infinitely many pairs of primes of the form $n,n+2$. This is the $k=2$ case of the prime-tuple prediction…

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,