80 problems

Minimal-polynomial degree conjecture. The degree of a minimal polynomial for the modulus $3^\gamma$ is $2d$.

Modular product representation conjecture. For sufficiently large $p$, there exist integers $x$ and $y$ such that

The classical multiplication-table conjecture. Every nonzero residue class modulo $p$ is representable as $xy\pmod p$.

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 $k\in\{1,\ldots,n\}$,…

Let $r=l/d$ with $\gcd(l,d)=1$ and $|r|>1$. Let $h_r:\mathbb{Z}\to\mathbb{Z}$ be a function in the class $\mathcal{P}_r$ of periodically linear maps … where $l_b\equiv-lb\pmod d$.…

Modular-3 conjecture. For every $k$,

Let $m$ be a positive integer, let $q=2^j\geq 4$, and let $A$ be a $q$-modular witness. Let the affine lift criterion refer to the criterion stated earlier in the paper for passing…

Let $m$ be a positive integer, and let a modular witness be a vertex set satisfying the corresponding modularity condition. Finite first-bit obstruction elimination. There exists a…

Let a modular witness be a vertex set satisfying the corresponding modularity condition, and let $m$ be a positive integer. For powers of two $q$, write $q$-modular for modularity…

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…

Conjecture on $E_4$. The currently established bound $E_4(r,j,M,H)$ in equation can be improved. The passage provides the expected order of magnitude above as motivation, with the…

For integers $p\geq q\geq0$, set … and let $T_{p,q}$ be the associated mapping. A triplet is strongly admissible when it has finitely many cycles and no divergent trajectory. Gener…

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…

Greaves–Yatsyna's sharpness conjecture. For every fixed $e\geqslant 3$ and all sufficiently large $n$, this bound is sharp.

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…

Sparse-edge conjecture. The number of edges in $T(n)$ is $o(n^2)$.

Let $a\in\mathbb{N}-\{0,1\}$ be an integer that is not a multiple of $10$. Let $\operatorname{APS}(a)$ denote the asymptotic phase shift of the tetration base $a$, represented as a…

Let $(V_n(p,q))_{n\geq 0}$ be the Lucas sequence considered in the paper, and let $\pi_V(m)$ denote its period modulo $m$. The vertical slice result refers to the additive-inverse…

Let $U_n:=U_n(p,q)$, let $m\geq 1$, and let $n\in\mathbb{Z}$. Assume that both $p$ and $m$ are even. If … then exactly one of the following conclusions occurs:…

Let $P=x^2y^2+(x^2+x+1)y+x^2$, and let $\theta$ be its Furstenberg series. For each $\alpha\geq 1$, consider the orbit of $\theta$ under the operator $\lambda_{0,0}$ modulo…

A modular polynomial on $\mathcal{GF}(p)$ in two variables $(n_1,n_2)$ is said to satisfy the weak generalization hypothesis when it has the form … where $g_1$, $g_2$, and $h$ are…

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.…

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…