36 problems

Generating-function conjecture. The dimensions of these spaces satisfy

Let $A=3$ and let $n\geq 0$ be an integer. Define … For a polynomial $P(K_1,K_2,K_3)$ in any of the four families specified in Theorem … sum{k1geq k2geq k3geq 1}frac{P(K1,K2,K3)}{(…

Let $n\geq 0$ be an integer. Let $\sigma\in\{2,3\}$ be the element satisfying $\sigma\equiv n\pmod 2$, and let $\widetilde{\sigma}$ be the other element of $\{2,3\}$. Let…

Let $a$, $b$, $c$, $x$ and $y$ be arbitrary integers. Consider a hexagon with side lengths $2n+a$, $2n+b$, $2n+c$, $2n+a$, $2n+b$, and $2n+c$, and a horizontal rhombus whose bottom…

Depth-one reduction conjecture. Every derivative sum in the $f_6$ family belongs to the subalgebra of the MZV algebra generated by the single zeta values $\zeta(m)$; no irreducible…

Let $S_1,S_2,S_3$ be the three summands defined from the WZ identities in the preceding construction, and let $V_N(S_i)$ denote the span of the coefficients of degree-$N$ parameter…

Image Hilbert–Poincaré conjecture. The generating function of the dimensions of the images is rational, with initial expansion

Hilbert–Poincaré rationality conjecture. The Hilbert–Poincaré series of these coefficient spaces is rational:

Let … be the hypergeometric value used in the paper. Irrationality conjecture. … This mild irrationality assumption is introduced for class 8, where the relevant even twist is not…

Let series 33 in Table have $t=-1$, and let $\mathcal{R}_n$ and $\mathcal{F}_n$ be the quantities associated with this series. Let $c_j$ be the coefficients defined by Theorem. Env…

Kresch–Tamvakis conjecture. The following terminating hypergeometric series has absolute value at most $1$:

Let $r\leq 1$ be an odd integer coprime with $5$, and let $p$ be an odd prime such that $p\equiv r\pmod{5}$ and $p\geq (5-3r)/2$. Then the proposed supercongruence. … This is prese…

Integrality conjecture. Then

Let $(x)_k$ denote the rising factorial, with $(x)_0=1$. Let $d\geqslant 2$ be an even integer, and let $p\equiv d+1\pmod{2d}$ be a prime. The hypergeometric supercongruence. … Thi…

Let $p\equiv 5\pmod{8}$ be a prime, and let $(a)_k=a(a+1)\cdots(a+k-1)$ denote the rising factorial. The p-adic supercongruence conjecture. … This is presented as a stronger versio…

Hypergeometric contribution conjecture. The ordered multi-degree contributions satisfy

Let $p$ be an odd prime and let $r$ be a positive integer. Equality conjecture. One should have … The conjecture is proposed because both sides are predicted to have the same Berno…

Let $(x)_0=1$ and $(x)_k=x(x+1)\cdots(x+k-1)$ denote the Pochhammer symbol. Guo–Schlosser's supercongruence. For every prime $p\equiv 2\pmod{3}$, one has … This is a supercongruenc…

Let $p$ be an odd prime, let $r$ be a positive integer, and let $\delta\in\{1,2\}$. Guillera–Zudilin-type conjecture. … The paper states that this stronger modulus cannot be proved…

Let $p>3$ be a prime, and let $r$ be a positive integer. The supercongruence. First, … More generally, … This conjecture concerns supercongruences modulo high prime powers and gene…

Let $r$ be a positive integer and let $p$ be a prime with $p>2r+1$. The supercongruence. … This is one of three proposed strengthenings related to the paper's earlier supercongruen…

Let $p\geq5$ be a prime with $p\equiv3\pmod{4}$, and let $\Gamma_p$ denote the $p$-adic Gamma function. Liu's conjecture. … The paper's abstract identifies this as a congruence con…

Let $p$ be an odd prime. For parameters $a_1,,a_6$ and $b_1,,b_5$, write the truncated hypergeometric series as … where $(a)_0=1$ and $(a)_k=a(a+1)\cdots(a+k-1)$ for $k\geq 1$. Let…

Let $p\geq 5$ be a prime satisfying $p\equiv 3\pmod{4}$, and let $\Gamma_p$ denote the $p$-adic Gamma function. A modulo $p^5$ extension of van Hamme's (A.2) supercongruence. … Thi…

Fix an integer $d\ge2$, let $p$ be an odd prime, and define … and … Let $\epsilon=\pm1$ and suppose that $F_p(\epsilon)$ is a $p$-adic unit. Higher-level supercongruence. For every…