18 problems

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 $p\ge7$ 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…

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

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

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, 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…

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 $G\in\mathcal{G}_n^{sc}$ 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 $\mathfrak{n}$ and $\mathfrak{m}$ be nonzero ideals chosen independently and uniformly at random from the set of ideals in $\mathbb{Z}[i]$ with norm at most $x$. Write…

Bradford–Ionascu conjecture. Such a decomposition exists with $a,b,c\in\mathbb{Z}[i]$ such that the real and imaginary parts of each of $a$, $b$, and $c$ are either both nonnegativ…

Gaussian Erdős–Moser conjecture. The equation above has only the solution $(k,m)=(2,3)$:

Asymptotic cardinality conjecture. The minimal cardinality satisfies