22 problems

Genčev–Rucki conjecture. For every $r\in\mathbb{N}$ with $r>1$,

For an integer $n\geq 0$, define … Let $s_3(m)$ denote the sum of the base-$3$ digits of the nonnegative integer $m$. The cubic binomial-sum valuation conjecture. For every integer…

Let $r\in\mathbb{N}_0$ and $n\in\mathbb{N}$. For any reals $x,y$, let ${\bf p}_{r+1}:=(p_1,p_2,\ldots,p_{r+1})\in\mathbb{N}^{r+1}$ and…

Let $n,r\geq 1$, let … and let $\mathcal{S}_{(r,n)}(\ell)$ denote the generalized binomial sum defined in the paper. Extended non-integrality conjecture. If $\ell\in\mathcal{L}$, t…

Monotonicity conjecture. The sequence $a_p$ is decreasing in $p$. The preceding asymptotic result gives $a_p\to 0$ as $p\to\infty$, so this conjecture would describe the finite-$p$…

Let $p>3$ be a prime with $p\equiv3\pmod4$, and let $r$ be a positive integer. The supercongruence. … This is described as a generalization of an earlier case distinction; the stat…

The supercongruence.

Central binomial sum conjecture.

Higher Apéry-limit conjecture. For $d\geq 5$, there is a unique such solution $C^{(d)}(n)$ satisfying

Apéry-limit conjecture. For $d\geq 3$, there is a unique such solution $B^{(d)}(n)$ satisfying

El Bachraoui's conjecture. For every odd prime $p$,

The first supercongruence conjecture. For every odd prime $p$,

Even-moment asymptotic conjecture. There is a constant $c(k)>0$ such that, as $j\to\infty$,

Let $m$ be a positive integer, and let $\gamma$ denote the Euler–Mascheroni constant. Wu's second asymptotic conjecture. … This is one of three asymptotic conjectures used to reduc…

Small-index multiplier conjecture. Set

Multiplier conjecture. There exist integers $a_{2r-1}$ and $b_r$, independent of $n$, such that

Let … and … with $s_{18}(0)=1$. Cooper's conjecture. For every prime $p\geqslant3$, … and, for every prime $p$, … These congruences are sporadic-sequence analogues of known congrue…

Let $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. For positive integers $r,s,t$ satisfying $r+s+t\equiv 1\pmod 2$, triple-product divisibility conjecture. The…

Let $B_{m,k}$ and $B_{n,k}$ denote the Catalan-triangle quantities used in the paper. Let $m,n,s,t\geqslant 1$ and $r\geqslant 0$, with $r+s+t\equiv 1\pmod 2$. Mixed-product divisi…

Let $n,r\geqslant 1$. Define $\alpha(N)$ to be the number of $1$'s in the binary expansion of $N$. Binary-weight divisibility conjecture. The sum … is divisible by … The conjecture…

Guo–Jouhet–Zeng's divisibility conjecture. For all positive integers $r,s,t,n$,

For positive integers $m$ and $n$, define … Here $\gcd(a_1,a_2,\ldots)$ denotes the greatest common divisor of the indicated integers. Guo–Jouhet–Zeng's gcd conjecture. … Calkin's…