65 problems

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

Let $E\geq 2$ and $n\geq 0$ be integers, and define … where … Also let $\operatorname{d}_n=\operatorname{lcm}\{1,2,\ldots,n\}$. Vasilyev's conjecture. For all integers $E\geq 2$ an…

Let a homogeneous space, such as a Grassmannian manifold, and simpler homogeneous spaces, such as products of projective spaces, have associated hypergeometric series; twisted hype…

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,…

For a positive integer $N$, let $q_{N,(N)}(z)$ denote the mirror-map series used in the paper, let $\Xi_N$ be the integer defined from the $p$-adic valuations of $H_N$, and let…

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…

A multiple hypergeometric series is called totally elliptic when the ratios of its successive coefficients are elliptic in the summation variables and parameters. Modular invarianc…

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…

Let $X$ be a Calabi–Yau variety, let $J_1,\ldots,J_r$ be suitable integral (semi-)ample generators of $H^{1,1}(X)\cap H^2(X,\mathbf Z)$, and let $w(x;J/(2\pi i))$ be the associated…

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

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…

Define the power series … and … Z.-W. Sun's conjecture. The nine displayed special values of $U$ and $V$ equal the stated algebraic multiples of $1/\pi$. These identities are part…

For $n\geq 0$, let $(a)_n$ denote the rising factorial, $(a)_n=a(a+1)\cdots(a+n-1)$, with $(a)_0=1$. Cullen–Zhao conjecture. The following two series identities hold: … These are R…

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…

Z.W. Sun's conjectured identity. The series equals

Y. Zhao's conjectured identity. The series equals

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…

For integers $d\geq2$ and $n$ satisfying $n\equiv-1\pmod d$ and $n\geq2d-1$, define the rising factorial by $(a)_k=a(a+1)\cdots(a+k-1)$ for $k\geq1$ and $(a)_0=1$. Wang–Li–Tang's i…

Long's conjecture. For every odd prime $p$, this sum satisfies

Deines et al.'s supercongruence. The congruence