42 problems

Let $a,n\in\mathbb{Z}^+$ satisfy $a>2$, $\left\lfloor\log_2(a)\right\rfloor+1\geq n>1$, and suppose there is no $k\in\mathbb{Z}^+$ such that $k^n=a$. Integer-part root conjecture.…

Let $n$ be a positive integer, and let $C_G(\mathbb{Z}_n)$ denote the number of isomorphism classes of generalized Weierstrass elliptic curves over $\mathbb{Z}_n$. Generalized Weie…

Fix $k\geq 2$, $m\geq 1$, and $0\leq a<m$. Set … Let $h_{m,a}^{(k)}(n)$ be the number of coefficients of $I_n^{(k)}(x)$ congruent to $a$ modulo $m$, and define … Congruence-count r…

Fix $k\geq 2$ and define $H_{m,a}^{(k)}(x)$ as the generating function counting coefficients of $\prod_{i=1}^n(1+x^{F_{i+k-1}^{(k)}})$ that are congruent to $a$ modulo $m$. Parity-…

Modulo-three conjecture. The sequence $1^n$ is removable if and only if

Let $q$ and $n$ be positive integers, and let $p_1,\ldots,p_n$ be distinct integers relatively prime to $q$ such that $q>(7/4)^n$ and … Assume that for every $k\in\{1,\ldots,n\}$,…

Modular-3 conjecture. For every $k$,

For powers-of-two moduli $2^k$, consider the modular Ackermann map $A_{2^k}(m,n)$ and its output distribution on $\mathbb Z_{2^k}$ as the recursion level $m$ varies. Hierarchy mixi…

For fixed $m\ge3$, consider the map … from uniformly chosen $n\in\mathbb Z_{2^k}$ to $\mathbb Z_{2^k}$. Asymptotic equidistribution conjecture. As $k\to\infty$, this map approaches…

Let $n$ be a positive integer, and let $\mathbb{Z}^{\widetilde{O}(n^{n^{1/3}})}$ denote the integers in the indicated size range. The residue-class formulation. There exists a set…

Let $[n]=\{1,2,\ldots,n\}$, let $\mathscr{P}(n)$ be the power set of $[n]$, and let a family be an $(a,b)$-town modulo $k$ when all its sets have cardinality congruent to $a$ modul…

For $k\in\mathbb{N}$, let $a_1,\ldots,a_k\in\mathbb{Z}/(2^r\mathbb{Z})$, and let $d\leq r$ be the largest non-negative integer such that $2^d$ divides $\sum_{i\in I}a_i$ for some n…

Generalized Weierstrass counting conjecture.

Let $n=p_1^{e_1}p_2^{e_2}\ldots p_l^{e_l}$ be an odd composite integer with $e_i\geq 1$, and let $C_R(\mathbb{Z}_n)$ denote the number of isomorphism classes of reduced Weierstrass…

Let $p$ be an odd prime and let $m$ be a positive integer with $\gcd(p^m,6)=1$. Let $\Delta^i(p^m)$ denote the number of solutions to $\Delta \equiv i \pmod {p^m}$, and let…

Let $p$ be prime, let $m$ be a positive integer, and write $\mathbb{Z}_{p^m}$ for the integers modulo $p^m$. Let $CR$ denote a cubic residue and let $[i]$ denote the $i$th class in…

Fix an even modulus $n\geq2$, a subset $S\subset\{0,\dots,n-1\}$ of residue classes, and the associated discrete stick-fragmentation process with stopping set determined by $S$. Le…

Let $S(n)$ denote the Schur-type overpartition function considered in the paper, and let $n$ be a nonnegative integer. Modulo 32 congruences. The following congruences are supporte…

Let $t\geq 0$ be fixed and let $n$ be sufficiently large. Set $N=2^n+t$, and let $A$ be an $n$-element set of positive integers whose subset sums are all distinct modulo $N$. Modul…

Maximum-period conjecture. When $\mu\bmod 3=1$, the maximum period is

Let $e$ be a positive integer, and let $S\subseteq\{0,1,\ldots,e-1\}$. Assume that for every $x,y\in S$, either $x+y\bmod e\in S$ or $x-y\bmod e\in S$. The odd-subset symmetry conj…

Let $\mathbb{Z}_n$ be the cyclic group of integers modulo $n$. An Erdős-deep family is a family whose distance multiset has multiplicities precisely $1,2,\dots,k-1$ for some intege…

Let $m$ and $n$ be positive integers with $m\mid n$, and let $a<b<\min\{m,n\}$. Consider the random walk on $\mathbb{Z}_m\times\mathbb{Z}_n$ whose moves are $(+a,+b)$ and $(+b,+a)$…

Let $r(x)$ be the generating function of the Rueppel sequence, and let the Hankel transform be reduced termwise modulo $2$. Modulo-2 periodicity conjecture. The Hankel transform of…

Let $\rho$ be a prime and let $M_\rho=2^\rho-1$. The Mersenne-prime residue conjecture. If $M_\rho$ is a Mersenne prime, then exactly one of the following holds:…