10 problems

Let $N$ be a prime congruent to $1$ modulo $4$, and let $\beta_{-4N}$ and $\beta_{-4}$ be the associated integers. Parity-matching conjecture. If … then $\beta_{-4N}$ and…

Let $E$ be an elliptic curve with conductor $N_E$, and let $\beta_{D}$ denote the coefficient associated with a fundamental discriminant or its square multiple as in the paper. Squ…

Let $E$ be an elliptic curve with conductor $N_E$, and let $a_E(2)$ denote its eigenvalue at $2$. For a prime $N\equiv 3\pmod{4}$, let $\beta_{-N}$ be the associated integer. Parit…

Let $P(n,m)$ denote the number of partitions of $n$ with $m$ missing integers. Define … Thus, $\mathcal{M}_e(n)$ and $\mathcal{M}_o(n)$ count partitions of $n$ with an even and odd…

Let $r\textgreater 1$ and let ${\boldsymbol{\sl{k}}}=(k_1,\ldots,k_r)\in(\mathbb{Z}_{>0})^r$ with $k_r>1$. The Borwein–Girgensohn parity conjecture. The multiple zeta value…

Let $K_1$ and $K_2$ be representatives of virtual knots $[K_1]$ and $[K_2]$, and choose any connected sum $K_1\#K_2$. Let $upr$ denote Manturov's parity group projection from virtu…

Let $g(q)$ be a mock theta function with integer coefficients, and let $c(g;n)$ denote the coefficient of $q^n$ in its series expansion. Say that $g$ is of parity type…

Let $a(n)$ denote the number of partitions of $n$ in which every part occurs with odd multiplicity. For a sequence, say that its odd values have density $d$ when the corresponding…

Let $L$ be a Latin square of even order $n$. Let $w$, $x$, $y$, and $z$ be the numbers of transversals in $L$ of types $T^{000}$, $T^{011}$, $T^{101}$, and $T^{110}$, respectively.…

Judge–Keith–Zanello conjecture. The density $\delta_t$ exists and equals $1/2$ for every positive odd integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq1$ odd, then $\delta_t$…