21 problems

Let $ped(n)$ denote the number of partitions of $n$ in which even parts are distinct and odd parts are unrestricted. Let $p\geq5$ be a prime satisfying … and let $t$ be a positive…

Let $\mathbb{K}$ be a finite field of characteristic different from $2$, and let $\mathbb{K}^{*}$ denote its multiplicative group. An element $x$ is a generator if it generates…

Let $p$ be an odd prime, and write $(\mathbb{Z}/p\mathbb{Z})^{*}$ for the multiplicative group of nonzero residue classes modulo $p$. A residue class $\overline{k}$ is a generator…

Kuperberg–Lalin's symplectic prime variance conjecture. For $x^{1/k+\varepsilon}\leq y$, one has

Subpandigital and subpenholodigital square existence conjecture. Suppose $b>7$. A strict subpandigital square and a strict subpenholodigital square in base $b$ exist if and only if…

For positive integers $m,n$ with $n$ odd, define … For each positive odd integer $m$, let … Finiteness conjecture. For every positive odd integer $m$, the set $E(m)$ is finite. In…

For positive integers $m,n$ with $n$ odd, define … For primes $p\equiv1\pmod4$, this determinant is an integer square, so choose a square root $\sqrt{D_p^{(3)}}$. Sun's quadratic-c…

Let $n$ range over positive integers such that $p=4n-1$ is prime, and let $J_n$ be the quantity defined earlier in the paper. Asymptotic conjecture for $J_n$. … The paper presents…

Maximum VC-dimension conjecture. One has

Let $D_p(c,d)$ denote the determinant associated with the quadratic form $i^2+cij+dj^2$ in the paper. Legendre-symbol formula. For any odd prime $p\equiv\pm2\pmod5$, … The supplied…

Let $p$ be a prime, and let $n(p)$ denote the least natural number that is not a quadratic residue modulo $p$. For any fixed $\epsilon>0$, the least quadratic nonresidue satisfies…

Sun's conjecture. If $d$ is non-zero and $c^2-4d\ne0$, then $[c,d]_p=0$ for infinitely many primes $p$.

Let $p$ be a prime, and let $n_p$ denote the least positive quadratic nonresidue modulo $p$. Here $f(p)\ll g(p)$ means that there is a constant $C>0$ such that $f(p)\leq Cg(p)$ for…

Let $0<\alpha\leq 1$ be a real number. Let $\mathbb{F}_p$ denote the finite field with $p$ elements, and let $\chi$ be the quadratic character modulo $p$. Chowla's conjecture. Ther…

Let $p$ be prime, let $\chi(x)=\left(\frac{x}{p}\right)$ denote the Legendre symbol, and let $\{r_1,\ldots,r_k\}\subseteq\mathbb{Z}_p$. For distinct indices $i,j$, set $a=r_j-r_i$…

Small quadratic non-residue conjecture. There exists a prime $q<2\sqrt{p}$ such that

The cubic-root product conjecture.

Let $p$ be an odd prime, let $g$ be a primitive root modulo $p$, and define $\sigma_g$ on $\mathbb Z/p\mathbb Z=\{0,1,\ldots,p-1\}$ by … for $b\in\{1,\ldots,p-1\}$, with…

General-level degree formula conjecture.

Let $d$ range over square-free positive integers, and let $\left(\frac{d}{p}\right)$ denote the Legendre symbol for an odd prime $p$. Quadratic-residue conjecture. All but finitely…

Half-factorial residue distribution conjecture. This proportion approaches $\frac{1}{2}$ as $p \to \infty$.