23 problems

Let $p$ be a prime with $p>3$. Rodriguez-Villegas's conjecture. … The conjecture was later confirmed by Mortenson and generalized by Z.-W. Sun to a congruence modulo $p^3$, so it i…

Sun's conjecture.

Let $A_{n,k}$ be the ballot numbers … For all $r\in\mathbb{N}$ and $n,s\in\mathbb{Z}^+$ such that $r+s\equiv 1\pmod 2$, and with $\varepsilon=\pm1$, Guo–Zeng's ballot-number divisi…

Sun's stronger congruence conjectures. For every positive integer $n$, and for every prime $p$,

Infinitude conjecture. Infinitely many primes satisfy this congruence.

Hasler's structural infinitude conjecture. The equation $\binom{3n}{n} \equiv 3^n \pmod{n}$ has infinitely many solutions of the form $n=3^k\cdot p$, where $k$ is a positive intege…

Let $b,n\in\mathbb Z^{+}$ and let $p$ be a prime satisfying $p\equiv\pm1\pmod b$ and $\langle-1/b\rangle_p\equiv0\pmod 2$, where $\langle x\rangle_p$ denotes the least nonnegative…

Z.-W. Sun's stronger congruence conjecture.

Z.-W. Sun's conjecture. The polynomial

Z.-W. Sun's conjecture. The polynomial

Sun's conjecture. If $p\equiv1\pmod 3$ or $2\mid a$, then

Adamchuk's conjecture. For any prime $p\equiv1\pmod 3$,

Generalized divisibility conjecture. is congruent to zero modulo

Sun's Ramanujan-type series conjecture.

Let $p$ be a prime with $p\geq 5$, and let $n,r,l$ be integers for which the displayed sums and factorial denominator are defined. Then Sun–Davis congruence. … This is an integer c…

Let $n\geq 5$ be odd and set … For $k\geq 1$, define … The set $W_4$ is the set of Wolstenholme primes. McIntosh's conjecture. The sets of integers $W_2$ and the Wolstenholme prime…

Let $p$ be a prime, and let … or … A Wolstenholme prime is a prime satisfying ${2p-1\choose p-1}\equiv1\pmod{p^4}$. Meštrović's conjecture. Either of the preceding congruences for…

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…

Nonexistence conjecture. There are no composite numbers $p$ satisfying either of these congruences. The conjecture extends the paper's supercongruence results, which establish the…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $n,r,s\geqslant 1$ with $r\not\equiv s\pmod 2$. Consecutive-index product congruence conjecture. … This…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r,s\geqslant 1$. Even-exponent Catalan-triangle congruence conjecture. … This is a complementary…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, let $r\geqslant 0$ and $s\geqslant 1$, and assume $n\geqslant 4^s-1$. Power-of-four threshold conjecture. ……

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper, and let $n,r\geqslant 1$. Odd-power Catalan-triangle congruence conjecture. … if and only if $n=2^a-2^b$ for…