41 problems

The q-log-convexity conjecture. (i) The sequence $(\sum_{k=0}^n\binom nk^3q^k)_{n\geq8}$ is $q$-log-convex; for every integer $m\geq2$, the sequence…

Generalized Binomial Biroot Conjecture. The approximations converge according to

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…

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 $[i]_{p,q}$…

Catalan triangle Laurent polynomial and positivity conjecture. This expression is a Laurent polynomial in $q$, and it has non-negative integer coefficients when…

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…

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

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

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

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. $P_n(z)$…

Strong conjecture. The class $c_k\big(\Pol^d(\mathbb{C}^n)\big)$ is of the form $P(B_1,\dots,B_s)$, where $P$ is a polynomial. This strengthens the weak conjecture by proposing an…

Let … The two displayed series are conjectured to converge to the stated multiples of $1/\pi$. Sun's 1/pi series conjecture. … and … The source presents these as open conjectures m…

Let $G(N)$ be the number of positive integers $n eleq N$ such that the central binomial coefficient $\binom{2n}{n}$ is coprime to $105$. Pomerance's conjecture. For all sufficie…

Unbounded multiplicity conjecture. For each $N\in\mathbb{N}$ there is $d_N\in\mathbb{N}$ such that

The q-Sun binomial conjecture. For every positive integer $n$,

Fixed-difference finiteness conjecture. Given a fixed difference $d$, the number of near collisions with difference $d$ is finite.

Difference-one near-collision conjecture. There are no other near collisions with difference $1$ than those given above.

de Weger's conjecture. There are no other collisions than those given above.

Cusick–Li–Stănică-type conjecture. No perturbation of the form $\sigma_{n,k}+X_1$ is balanced except for the trivial cases above.

Let $m$ be a positive integer. Krattenthaler's conjecture. There exist positive integers $a$ and $b$ satisfying $am>b$ such that … for every integer $n\geq1$. This conjecture conce…

Fix integers $d$, $k\geq 0$, and $r\geq 1$. The coefficients $a^{(r)}_{d,k,j}$ are intended to expand powers of $\binom{n+dk}{k}$ in the family $\binom{n+dj}{j}$. Generalized expan…

Let $a,b$ and $n$ be positive integers with $a>b$. For integers $x$ and $y$, write $x\mid y$ when $x$ divides $y$. Amdeberhan–Moll's divisibility conjecture. … The conjecture conce…