32 problems

Let $A_{\mathcal K,k,m}(0)$ denote the constant term for the second family of meromorphic modular forms. K-family prime-parameter conjecture. The following assertions hold: if…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term, and let $d_3(k)$ be the sum of the base-3 digits of $k$. 3-adic valuation conjecture. If $m\equiv0\pmod3$,…

Let $d_2(k)$ be the sum of the binary digits of $k$. Two-digit valuation conjecture. If $d_2(k)=2$, $m=4j+2$ for $j=1,2,\ldots$, and $a=\operatorname{ord}_2(m)=1$, then … This is a…

Catalan valuation conjecture. Then $t=C_{1,n}$. The source presents this as an observed pattern in its empirical study of constant-term valuations; no resolution is given.

Let $d_2(k)$ be the sum of the binary digits of $k$, let $a=\operatorname{ord}_2(m)$, and let $o=\operatorname{ord}_2(A_{\overline{\mathcal K},k,m}(0))$. Binary-digit valuation con…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. Odd-prime valuation conjecture. If $p>2$ is prime and $a$ is a positive integer, then … This is an empiric…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. 2-adic valuation conjecture at $k=2$. For every integer $a\geq2$, … This is an empirical special case of t…

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

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…

Eswarathasan–Levine conjecture. There are infinitely many harmonic primes.

Eswarathasan–Levine conjecture. The set $J_p$ is finite for all primes $p$.

Let $C_n=\frac{1}{n+1}\binom{2n}{n}$ denote the $n$th Catalan number, and let $\nu_p(m)$ be the exponent of the prime $p$ in the prime factorization of $m$. Assume that $a,b\geq 1$…

Let $S(n,k)$ denote the number of partitions of an $n$-element set into exactly $k$ nonempty subsets, and let $v_2$ denote the $2$-adic valuation. For fixed $k\in\mathbb{N}$, consi…

Let $V$ be a vector subspace of $\bigoplus_{i=1}^\infty \mathbb{Q}$. For a prime $p$, let $\nu_p$ denote the $p$-adic valuation, and let $p_1,p_2,\dots$ be the increasing enumerati…

Let $\mathcal{ML}$ be the set of primes $p$ for which the Marques–Lengyel conjecture holds, and let $\mathcal{NMLR}$ be the set of primes for which the rational-center variant fail…

Let $p$ be a prime number, and let $T(n)$ be the Tribonacci sequence defined by $T(0)=0$, $T(1)=T(2)=1$, and $T(n+3)=T(n+2)+T(n+1)+T(n)$. Write $\nu_p(m)$ for the exponent of $p$ i…

Let $p$) be a prime number, and let $T(n)$ be the Tribonacci sequence defined by $T(0)=0$, $T(1)=T(2)=1$, and $T(n+3)=T(n+2)+T(n+1)+T(n)$. Write $up(m)$ for the exponent of $p$ in…

2-adic valuation conjecture.

Universal odd-prime valuation conjecture. Every $n\in\mathbb{N}$ satisfies condition (1) or (2). Consequently,

Let $S_{k,m}(n)$ be the $n$th coefficient in the power-series expansion defining the $m$-colored $k$-ary partition function, and let … Non-prime-power valuation conjectures. If $k$…

Let $S_{k,m}(n)$ be the $n$th coefficient in the power-series expansion defining the $m$-colored $k$-ary partition function, and define … 4-adic valuation conjectures. The followin…

Let $d_m(n)$ denote the $m$-colored $p$-ary partition function, let $p\in\mathbb{P}_{\geq3}$, and let $\nu_p$ denote the $p$-adic valuation. Write $m=pk+i$ with $k\in\mathbb{N}$ an…

Let $c_m(n)$ denote the $m$-colored binary partition function, and let $u2$ be the 2-adic valuation. For positive integers $k,n$ and nonnegative integers $k$ as specified below, c…

Let $A_{p,k}(n)$ denote the number of $k$-colored partitions of $n$ into powers of the prime $p$, and let $up$ be the $p$-adic valuation. Write $\mathbb{P}{\geq 3}$ for the primes a…

Let $t$ be an integer with $t\geq 2$, and let $n$ and $d$ be positive integers. For an element of ${\mathbb Z}/(t^n-1){\mathbb Z}$, write its standard $t$-ary expansion with digits…