16 problems

Let $p$ be an odd prime, and let $A_n$ denote the first kind of Apéry-like numbers used in the paper. Then Z.W. Sun's conjecture. … The conjecture was confirmed by Guo and Zeng; it…

Jacobi-symbol conjecture for $c_p$. For any odd prime $p$,

Anatomy-of-integers conjecture. For a fixed integer $s\geqslant 1$ and any integers $2\leqslant h<u$, we have

Sun's 2024 conjecture. If $p\equiv1\pmod4$, then

Let $p>3$ be a prime, and let $M_p^+$ be the matrix obtained from $[(\frac{i+j}{p})]_{0\leqslant i,j\leqslant (p-1)/2}$ by replacing every entry in the first row by $1$. Let…

Let $p>3$ be a prime. Let $A_p$ be the $(p-1)/2$ by $(p-1)/2$ matrix obtained from $[(\frac{i+j}{p})]_{1\leqslant i,j\leqslant (p-1)/2}$ by replacing its first row with…

Jacobi's conjecture. One has

Let $p>3$ be a prime, define … and, when $p\equiv3\pmod4$, let $h(-p)$ be the class number of $\mathbb{Q}(\sqrt{-p})$ and set $c_p=\left(2-\left(\frac{2}{p}\right)\right)h(-p)$. Th…

Let $p>p_0$ be prime, assume the distinct-factor bound conjecture and the uniform-distribution conjecture, and write … where $q_1<\cdots<q_k$ are the prime factors and…

Let $p\ge 3$ be prime, write $2^p-1=q_1\cdots q_k$ with distinct prime factors, and let $\mathcal A(p)$ be the selected set of odd integers $a\in[5,p-2]$ for which the associated p…

Let $p$ be a prime with $p\ne2,11$, let $A_n'$ be the sequence used in the paper, and let $\left(\frac{p}{11}\right)$ denote the Legendre symbol. If $\left(\frac{p}{11}\right)=1$,…

Let $p$ be an odd prime, let $0\leq\alpha\leq\frac{1}{2}$, and define … where $\left(\frac{n}{p}\right)$ is the Legendre symbol. Majority nonnegativity conjecture. For every…

Let $p$ be an odd prime and define … The author’s vanishing conjecture. If $p\equiv1\pmod 4$, then … The paper records this as an extension of the corresponding theorem for primes…

Let $p>3$ be a prime, and let $f_n=\binom{n}{0}^3+\binom{n}{1}^3+\cdots$

Let $p$ be an odd prime, and let $x$ and $y$ be integers satisfying $p=x^2+3y^2$ when $p\equiv1\pmod 3$. Write $\left(\frac{-1}{p}\right)$ for the Legendre symbol. Sun's conjecture…

Let $p$ be an odd prime, and let $E_p^+$ and $E_p^-$ be the convex $(p+1)$-near-edges associated respectively to the sequences of Legendre symbols and their negatives. Let…