10 problems

Pure reversed-prime asymptotic conjecture. For every $A>0$,

Let $b\geq 2$ be fixed. Define … and … Also let $\#\mathfrak{B}(N)$ denote the number of admissible integer representations encoded by the source's set $\mathfrak{B}(N)$. Asymptoti…

Nnameir hypothesis. For sufficiently large $N$ and $B\geq 1$,

Davenport's geometric-condition conjecture. If $C$ satisfies Davenport's Geometric Condition, then the asymptotic formula for the number of solutions to $C(\mathbf{x})=0$ holds wit…

Let $Q_1,\ldots,Q_n$ be a system of integral polynomial equations, and let $\operatorname{rk}_{\mathbb Q}(Q_1,\ldots,Q_n)$ denote its Schmidt rank over $\mathbb Q$. Cook–Magyar con…

For $N\in\mathbb{N}$, let $\mathbb{S}^1_+(N)$ be the upper semicircle of radius $\sqrt N$, and let $S_N=\mathbb{S}^1_+(N)\cap\mathbb{Z}^2$. Cilleruelo–Granville conjecture. For eve…

Hooley's conjecture. A similar phenomenon should hold in a circle method approach to counting integral solutions of

Let the forms under consideration have an $h$-invariant, and let Theorem … should hold assuming only that the $h$-invariant is sufficiently large. The conjecture proposes replacing…

Let $\mathcal A=\{n\in\mathbb N:n\equiv 4\pmod{24}\}$. A representation as a sum of four squares of primes is an expression … with each $p_i$ prime. Four prime squares representati…

Let $\tilde{G}(k)$ be the smallest integer $s$ for which the circle-method asymptotic formula for the number $r_{s,k}(l)$ of representations of $l$ as a sum of $s$ positive $k$th p…