111 problems

Let $P_n(z)$ be the sorted binomial polynomial with parameter $n$. Irreducibility conjecture. $P_n(z)$ is irreducible over $\mathbb{Q}$ for all even $n$. The analogous factorizatio…

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…

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

Singmaster's conjecture. The function $N(a)$ is bounded:

de Weger's conjecture. Apart from trivial equalities, every equality of binomial coefficients belongs to the infinite family discovered by Lind and Singmaster, with the exception o…

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 $n$ range over the positive integers, and write $\binom{2n}{n}$ for the central binomial coefficient. Pomerance's conjecture. The divisibility condition … holds on a set of pos…

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

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…

Sun's conjecture. If

For a positive integer $r$, let $b_k(r)=\binom{k+r-1}{r}$, and let $R_n(b_k(r))$ denote the corresponding reciprocal Hankel matrix. Integrality criterion conjecture. The inverse of…

Consider the triangular array with entries $\binom{n}{k}^2$ for $0\leq k\leq n$. Conditions (C1) and (C2) are the two conditions imposed on a triangular array in the source's theor…

Digit-statistics gcd conjecture. For every positive integer $n$,

The alternating-sum gcd conjecture. For all positive $m$ and $n$,

The 6n divisibility conjecture. For all positive $r$, $s$, $t$, and $n$, $S_{6}(r,s,t;n)$ is divisible by both

Alpha-one extension of the generalized Lucas congruence. The congruence (1.8), equivalently (1.7), also holds when $\alpha=1$.

Shifted Pascal-matrix conjecture. Its characteristic polynomial satisfies

Let $m$ and $k$ be nonnegative integers, let $p$ and $q$ be parameters, and let $Z_p$ be the generator appearing in the $(p,q)$-deformed generalized Weyl algebra. Write $[i]_{p,q}$…

Erdős's valuation growth conjecture. The $p$-adic valuation of the central binomial coefficient satisfies

Let $m$ be an odd integer. Eventual divisibility conjecture. There is an $N\in\mathbb{N}$ such that … for every $k\geq N$. This strengthens the preceding central-binomial-coefficie…

Let $n$ be an integer greater than $4$, and consider the central binomial coefficient $\binom{2n}{n}$. Divisibility conjecture. The central binomial coefficient $\binom{2n}{n}$ is…

Let $P_n(z)$ be the sorted binomial polynomial. For odd $n=2m+1$, the source identifies the relevant target group as the $m$th hyperoctahedral group. Hyperoctahedral Galois group c…

Let $P_n(z)$ be the sorted binomial polynomial, and let $q_n(z)$ denote the center truncation. Full Galois group conjecture. It is conjectured that $q_n(z)$, and all nontrivial tru…

Let $P_n(z)$ denote the sorted binomial polynomial, and call a root nontrivial when it is not the trivial root shared by the relevant polynomials. Common-root conjecture. $P_n(z)$…