32 problems

Wolstenholme's higher-power congruences. For every such $k$ and $p$,

Non-square conjecture. For every $n\ge3$, both $b_n^+(c)$ and $b_n^-(c)$ are non-square in $\mathbb{Z}[i]$.

Second-iterate irreducibility conjecture. If $f_r^2(x)$ is irreducible over $K$, then $f_r$ is stable over $K$.

Stable-factorization conjecture. The polynomial $f_r$ is eventually stable over $\mathbb{Q}(i)$ with constant $C(f_r,0)=4$, and exactly one of the following cases holds:

Let $\mathbf G_n(p)=\sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n$ and let $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. Explicit small-prime valuation conjecture. For $p=3$, every $n\ge1$ satisf…

Let $p\ge7$ be a prime with $p\equiv3\pmod4$, and write $n=r(p-1)$ with $r\ge1$. Odd-multiple valuation conjecture. … Equivalently, among multiples of $p-1$, … for $p\equiv3\pmod4$…

Let $p$ be an odd prime and define … with $v_p(x+yi):=\min\{v_p(x),v_p(y)\}$. The mod-4 valuation conjecture. For $1\le n\le p-2$, … and for $4\le n\le p-2$, … These formulas are s…

Hansel–Safer conjecture. Assuming the four exponentials conjecture, if $S$ is both $\alpha$- and $\beta$-recognizable, then $S$ is eventually periodic.

Lucas congruence for Gaussian binomial coefficients. For positive integers $A,B,C,D$ for which these coefficients are defined,

Conjectural coefficient pattern. If $p=4k+3>3$, then

Kuperberg–Lalin's Gaussian-direction variance conjecture. If $x\leq K^\ell$, then as $x\to\infty$,

Gaussian localization conjecture. If $\pi$ is not real, this ring is an extension of $\mathbb Q_{N(\pi)}$; if $\pi\equiv3\bmod4$ is a real prime, it is the ring of $2\times2$ matri…

Let $n$ be an even Gaussian integer, let $N(n)$ denote its norm, and let $\arg(n)$ denote its argument. Gaussian Goldbach conjecture with angular restrictions. Every even Gaussian…

Let $L$ be a primitive Gaussian line, meaning a line in the complex plane containing two, and hence infinitely many, coprime Gaussian integers. Let $g_L$ be the smallest positive i…

Let $L$ be a primitive Gaussian line, and let $\alpha_n$ denote the indexed Gaussian integers on $L$. For $\beta=x+iy\in\mathbb{Z}[i]$, write $N(\beta)$ for its Gaussian norm. Weak…

Let $L$ be a primitive Gaussian line, and let $\alpha_n$ denote the indexed Gaussian integers on $L$. For $\beta=x+iy\in\mathbb{Z}[i]$, define … Here $N$ is the Gaussian norm. Stro…

A Gaussian prime is a Gaussian integer that is prime in $\mathbb{Z}[i]$. A bounded-length walk to infinity through Gaussian primes is a sequence of Gaussian primes starting at the…

Three-prime sector conjecture. For every $z\in\Gamma_G$ with $\max(\operatorname{Re}(z),\operatorname{Im}(z))\geq6$, $z$ is a sum of at most three odd Gaussian primes…

Bounded odd Gaussian-prime representation conjecture. There are constants $c_0$ and $k_0$ such that, for every $z\in\Gamma_G$ with…

Let $A=\{z\in\mathbb{Z}[i]\mid\operatorname{Re}(z)>0\text{ and }\operatorname{Im}(z)>0\}$, and let…

Let $a,b$ be positive integers with $a>6$ and $a\geq b$. For a positive integer $k$, consider nonnegative integers $x_{1s},x_{2s}$ satisfying … and, for every $s$,…

Let $G\in\mathcal{G}_n^{sc}$ be a self-converse mixed graph, let $W=W(G)$ be its walk-matrix, let $\mathscr{U}_G$ be the associated set of Gaussian rational unitary matrices, and l…

Let $S=R=\mathbb{Z}[i]$, and let $(l_1,l_2,\ldots)$ be the sequence of lengths associated with a fixed divisor sequence, where $l_k$ is the smallest number of initial elements dete…

Let $n$ be a Gaussian integer with norm $N(n)>10$, and let $p,q$ be Gaussian primes. Conjecture F. One can write $n=p+q$ so that the angles between $n$ and $p$, and between $n$ and…

Let $Q$ denote the relevant set of Gaussian integers, and let $k+ki$ be a Gaussian diagonal integer with $k>1$. A Gaussian diagonal prime is a Gaussian prime of the form $a+ib$ in…