52 problems

Let $n$ be a positive integer, let $q$ be an indeterminate, let $\Phi_n(q)$ be the $n$th cyclotomic polynomial, and let $\left(\frac{-3}{n}\right)$ denote the Jacobi symbol. Guo's…

Sun's divisibility conjecture. Sun conjectured that

Central binomial sum conjecture.

Strazdins's conjecture. There is a monic polynomial $\tilde{P}_k(x)$ such that

Let … The values $S_1(n)=n\binom{2n}{n}$ suggest the following claim. Odd absolute-binomial sum conjecture. For every relevant nonnegative integer $r$, $S_{2r+1}(n)$ equals…

Conjecture 1.2. If $p\mid n$ or $p-1\nmid n-1$, then

Monotonicity conjecture. The sequence

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 $V_n$ be the Apéry-like sequence … For a prime $p>3$, the first author's conjecture asserts First author's conjecture. … The case $p\equiv1\pmod4$ is verified in the paper, whe…

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 … Here $n$ is a positive integer. Zhi-Wei Sun's conjecture. The number $a(n)$ is a positive odd integer for every $n>0$. The paper states that this conjecture is proved using t…

Bala's conjecture. The sequence also satisfies

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.

Luca–Pomerance nonintegrality conjecture. The sum $S_r(n)$ is never an integer.

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

Franel's conjecture. The sequence $A^{(d)}(n)$ satisfies a linear recurrence of order $\lfloor (d+1)/2\rfloor$ with polynomial coefficients.

Let $\lambda_1$ and $\lambda_2$ be two distinct integers. Carnevale–Voll conjecture. … This conjecture concerns the absence of unitary factors in polynomials arising in the study o…

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

Let … Here $\operatorname{Li}_4$ and $\operatorname{Li}_{3,1}$ denote single and multiple polylogarithms, respectively. Borwein's conjecture. The sum $S$ is expressible via…

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

Let $n,l\in\mathbb Z^+=\{1,2,3,\ldots\}$ and let $x$ be an indeterminate. Define … A polynomial $P(x)\in\mathbb Q[x]$ is integer-valued when $P(a)\in\mathbb Z$ for every…