335 problems

Let $c(n)$ denote the coefficient sequence under consideration, and let $n\in\mathbb{N}_0$. The congruence conjecture for $c(32n+23)$. … This conjecture is proposed as an open ques…

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

Harder's conjecture. If $p_{\mathfrak{p}}>2k+j-2$ and $\mathfrak{p}\mid L^{\operatorname{alg}}(k+j,f)$, then there exist a Hecke cusp eigenform

Let a super-Wolstenholme prime be a prime satisfying the Wolstenholme-prime congruence modulo a higher power of $p$. McIntosh's conjecture. No super-Wolstenholme primes exist. The…

For an integer $n>2$, define the left factorial by … Kurepa's conjecture. One has … equivalently, for every odd prime $p>3$, … This is a classical open problem concerning the arith…

For an integer $n>1$, define … Giuga's conjecture. The integer $n$ is prime if and only if … This is a power-sum characterization of primality, presented in the source as Giuga's c…

For a positive integer $n$, let $\overline{p}(n)$ denote the number of overpartitions of $n$. A Ramanujan-type congruence is a congruence of the form…

Let $M$ be a positive integer, and let $p(n)$ denote the number of partitions of the positive integer $n$. For an integer $r$, consider the residue class of $p(n)$ modulo $M$. Newm…

Sun's divisibility conjecture. Sun conjectured that

Wolstenholme's higher-power congruences. For every such $k$ and $p$,

Let $C_k$ denote the $k$th Catalan number, and let $p>3$ be a prime. Sun's Catalan congruences. … and … The paper states that these two conjectures are confirmed by its theorem for…

Let $q>1$ be a power of $3$, and define … The quotient $S_q/q^2$ is an integer. Sun–Tauraso's conjecture. … Equivalently, the central binomial coefficient sum is congruent to $q^2$…

Let $p$ be an odd prime. Consider the residue classes modulo $p$ represented by the factorial sequence $n!$ as $n$ varies. Factorial residue-class omission conjecture. About $p/e$…

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. Das–Saikia–Sarma's second conjecture. For all $i\geq 1$, all $n\geq 0$, and integers…

Let $\overline{p}_{q}(n)$ denote the number of overpartition $q$-tuples of $n$, where $q$ is prime. Saikia's conjecture. For all $n\geq 0$ and primes $q$, … These congruences exten…

Let $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k$ be the Apéry numbers of the second kind. For an odd prime $p$, if $p=x^2+4y^2$ with $x,y\in\mathbb Z$, then…

Let $p$ be a prime and let $E$ and $E'$ be elliptic curves over $\mathbb{Q}$. They are $p$-congruent if their $p$-torsion subgroups are isomorphic as Galois modules. Fisher's conje…

Let $f$ be a cusp form of weight $2$ and $g$ a cusp form of weight $4$ for $\Gamma_0(p)$, and suppose that there is a mod-$p$ congruence between $g$ and the derivative of $f$. Let…

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…

Z. W. Sun's conjecture. The following congruences hold:

Sun's divisibility refinement. For every positive integer $n$,

Let $p(n)$ denote the partition function, and let $B\pmod{A}$ be any arithmetic progression. Subbarao's conjecture. There are infinitely many integers $N\equiv B\pmod{A}$ for which…

Let $p(n)$ denote the partition function, and let $B\pmod{A}$ be any arithmetic progression. Ahlgren–Ono conjecture. There are infinitely many integers $N\equiv B\pmod{A}$ such tha…

Let $S_m(n)=\sum_{j=1}^{n-1}j^m$, and let $k$ be a positive integer. Even-modulus power-sum conjecture. For every $k\ge 1$ and every even positive integer $n$, the congruence … hol…

Atkin–Swinnerton-Dyer basis conjecture. The Atkin–Swinnerton-Dyer basis is $\{H_1,H_2\}$ if $p\not\equiv 5 \mod 12$, and it is $\{H_1+H_2,H_1+(p^2+1)H_2\}$ when $p\equiv 5 \mod 12$…