53 problems

Let $p\equiv 1\pmod 4$ be a prime, and let … where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol. Write … with $c_p,d_p\in\mathbb{Z}$ and $c_p\equiv 1\pmod 4$. Cohen–Sun–Vs…

Sun's conjecture. The element $2B_p$ is a quadratic residue modulo $p$. This conjecture was later confirmed, so the claim is solved.

Sun's conjectures. The following assertions hold:

Let $p$ be a prime, and let the least quadratic non-residue modulo $p$ be the smallest positive integer that is not a quadratic residue modulo $p$. Linnik's conjecture. For every…

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…

Jacobi's conjecture. One has

Let $p$ be a prime, and let $\mathcal R_p$ denote the set of quadratic residues modulo $p$. Write $\mathcal R_p\cup\{0\}$ for the quadratic residues with $0$ included. Lev–Sonn con…

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 $p\equiv\pm1\bmod 8$ be prime, put $r=\lfloor\log_2 p\rfloor$, and let $B$ be a Boolean function in $r$ variables representing the Legendre symbol by … for…

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

Let $p$ be a prime with $p\equiv 5\pmod{6}$, and let … denote the determinant of the $(p-1)\times(p-1)$ matrix with $(i,j)$-entry $(i^2-ij+j^2)^{-1}$ modulo $p$. Sun's conjecture.…

Let $\mathcal{M}'$ be the set of completely multiplicative functions $f:\mathbb{N}\to\{+1,-1\}$. A function in $\mathcal{M}'$ has length $k$ if $k$ is the largest positive integer…

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…

Let $p>3$ be a prime, let $d\in\mathbb Z$ with $p\nmid d$, and define … Let $h(-p)$ be the class number of $\mathbb Q(\sqrt{-p})$. Wu–She–Wang formula. … This was conjectured by Zh…

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 $k\geq 3$ be the number of players, and let $f:(\{0,1\}^n)^k\rightarrow\{0,1\}$ be any function. A $k$-player private simultaneous messages protocol for $f$ is said to have com…

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…

For each odd prime $p$, define the Fekete polynomial … where $\left(\frac{a}{p}\right)$ is the Legendre symbol. Fekete's conjecture. The polynomial $F_p(x)$ has no real roots in th…

Let $p>3$ be a prime and let $d\in\mathbb{Z}$ with $p\nmid d$. Define … where $\left(\frac{\cdot}{p}\right)$ is the Legendre symbol, and let $h(-p)$ be the class number of…

Let $p$ be a prime greater than $13$. Discriminant-balancing conjecture. There exists an integer $a$ satisfying … such that $(-1)^{(p+1)/2}a(a-1)$ is a square modulo $p$. The paper…

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

Sun's conjecture. If $p\equiv\pm3\pmod8$, then $-T_p$ is a quadratic non-residue modulo $p$.