43 problems

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

Boyd's valuation-four conjecture. The inequality $\nu_p(H_n)\geq 4$ never occurs.

Let $K$ be a number field, let $p$ be a prime, and let $\mathcal{O}_K$ be its ring of integers. Write $\nu_p(I(K))$ for the largest integer $k$ such that $p^k\mid I(K)$, where…

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…

Gessel–Lengyel equality conjecture. Equality should always hold in both of these inequalities. The claim was conjectured by I. M. Gessel and T. Lengyel in 2001, but the present sou…

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…

Let $a(n)=\sum_{0\leq i\leq n}\binom{n}{i}\binom{n+i}{i}$ be the central Delannoy number, and let $b(n)=\nu_3(a(n))$. Shallit's central Delannoy valuation conjecture. The sequence…

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 $V$ 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…

Let $L(n)$, for $n\geq 0$, be the number of tilings of the Aztec diamond of order $n$ using horizontal skew tetrominos and square tetrominos. Write the multiplicity of 2 in the pri…

2-adic valuation conjecture.

The 2-adic limiting valuation conjecture. For even $p$ and $\alpha\leq v_2(d)+1$, one has

Consider the numeration system of Example, and let $\alpha_1$ and $\alpha_2$ denote the exponents governing the growth of the $2$-adic and $3$-adic valuations $\nu_2(U_i)$ and…

Consider the numeration system of Example, and let $\alpha_1$ denote the exponent governing the growth of the $2$-adic valuations $\nu_2(U_i)$ in Lemma. Conjecture on the valuation…

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 $c_m(n)$ be the $m$-colored binary partition function and define, for $m\in\mathbb{N}_{+}$, … Boundedness conjecture. If $k\in\mathbb{N}_{+}$ and $m=2^{2k}-1$, then the sequenc…

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 $\nu_p$ be the $p$-adic valuation. Let $\mathbb{P}$ denote the set of prim…