150 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…

Let $p$ be an odd prime, and for $n\geq 0$ let $(a)_n=a(a+1)\cdots(a+n-1)$ denote the Pochhammer symbol. Van Hamme's (B.2) supercongruence. … This is the stated $p$-adic analogue o…

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…

Z.-W. Sun's conjecture. For every prime $p$ satisfying $p\equiv1\pmod3$,

Beukers' conjecture.

Sun's conjecture.

Let $n\in\mathbb N=\{0,1,2,\ldots\}$ and define the Apéry-like numbers … For an odd prime $p$, let $\left(\frac{\cdot}{p}\right)$ denote the Legendre symbol. Kimoto–Wakayama superc…

Swisher's general VanHamme-type supercongruence conjecture (F.3).

Let $D_n$ and $S_n$ denote the Delannoy and Schröder numbers, respectively, defined by … Let $p>3$ be a prime. Sun's supercongruence. … This congruence was raised by Z.-W. Sun in 2…

Let $\mathbb{N}^+$ denote the positive integers, let $a_i(n)$ be the sequence defined by … and let $p\geq 5$ be prime. The Almkvist-Zudilin supercongruence. For every…

Sun's binomial-cube congruence conjecture. The displayed congruence holds.

Let $r\in\mathbb{Z}^{+}$ and let $p>3$ be prime. Write $B_{p-3}$ for the Bernoulli number indexed by $p-3$. Sun's conjecture. … This extends the corresponding congruence modulo $p$…

Guo–Liu–Schlosser conjecture. Under the respective hypotheses above, the corresponding congruence holds. The modulus-$p^5$ restriction of the first congruence was confirmed by Guo,…

Let $T_k(b,c)$ denote the generalized central trinomial coefficient. For an odd prime $p$, let $\left(\frac{a}{p}\right)$ be the Legendre symbol. Sun's conjecture. … If $p>3$, then…

Let $n\geq 3$ be an integer, let $p$ be a prime satisfying $p\equiv 1\pmod n$, and let $\Gamma_p(x)$ denote the $p$-adic Gamma function introduced by Morita. Deines–Fuselier–Long–S…

The $1458$ supercongruence conjecture. One should have

Let $p$ be an odd prime. If $p=x^2+y^2$ with $x$ odd, define the corresponding integers $x,y$ as in the assertion; otherwise suppose $p\equiv 3\pmod 4$. Beukers' conjecture. … This…

Let $n$ be a positive odd integer with $n\equiv 3\pmod{4}$, let $q$ be an indeterminate, and let $(a;q)_k$ denote the $q$-shifted factorial. Write…

Let $p>3$ be a prime, and let $\legendre{-6}{p}$ denote the Legendre symbol of $-6$ modulo $p$. Sun's supercongruence. One has … This is a $p$-adic analogue of the preceding series…

Let $\mathbf G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n$ and let $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. Explicit small-prime valuation conjecture. For $p=3$, every $n\ge1$ satisf…

Let $p\ge7$ be a prime with $p\equiv3\pmod4$, and write $n=r(p-1)$ with $r\ge1$. Odd-multiple valuation conjecture. … Equivalently, among multiples of $p-1$, … for $p\equiv3\pmod4$…

Let $p$ be an odd prime and define … with $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. The mod-4 valuation conjecture. For $1\le n\le p-2$, … and for $4\le n\le p-2$, … These formulas are s…

Let $C/\mathbb{Q}$ be an elliptic curve with $j(C)\notin\{0,1728\}$, and for $k\in\{4,6,8,10,14\}$ and $1\leq r\leq k-1$ write … Let $p$ be a supersingular prime of $C$ with…

Let $C/\mathbb{Q}$ be an elliptic curve with $j(C)\notin\{0,1728\}$. A prime $p$ is supersingular for $C$ when $a_p(C)=0$ (for $p\geq5$), where $a_p(C)=p+1-|C(\mathbb{F}_p)|$. Let…

For $n\in\mathbb N$, define … Let $p$ be an odd prime. Sun's weighted supercongruences. … … If $p>3$, then … … These congruences are presented as a conjecture of Sun; the supplied…