31 problems

Let $p(n)$ denote the number of partitions of the positive integer $n$. Parkin–Shanks conjecture. The values $p(n)$ that are even and the values $p(n)$ that are odd each have natur…

Let $G$ be a regular graph. A spanning even tree of $G$ is a spanning tree in which every vertex has even degree. Jackson–Yoshimoto conjecture. Every connected non-bipartite regula…

Judge–Keith–Zanello conjecture. For every odd positive integer $t$, $\delta_t=1/2$.

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 $a_E(2)$ denote its eigenvalue at $2$. For a prime $N\equiv 3\pmod{4}$, let $\beta_{-N}$ be the associated integer. Parit…

Birch and Swinnerton-Dyer parity prediction. Integers $n\equiv 5,6,7\pmod 8$ should be congruent numbers.

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 $b_6(n)$ denote the number of 6-regular partitions of $n$, that is, partitions whose parts are not divisible by $6$. Equidistribution conjecture. The value $b_6(n)$ is odd for…

Quadratic-progression parity conjecture. The theorem that both parities occur infinitely often among these values, with $(m,6)=1$, should hold for all square-free $D$ satisfying…

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…

Partition Parity Conjecture. The values $p(n)$ are distributed equally between odd and even values, namely

Half-oddness conjecture. When $n\equiv 7\pmod{8}$, “half” of the values of $c_1(n)$ are odd.

2-parity conjecture.

Let … where $\operatorname{A001850}$ and $\operatorname{A001003}$ denote the sequences identified by those OEIS entries. Zhi-Wei Sun's conjecture. The number $a(n)$ is an odd integ…

For odd $k$ among $5,7,9,13,15,17,21$, let $\overline{C}_{4k,k}(n)$ denote the corresponding $(4k,k)$-singular overpartition function. Parity-density conjecture. The values…

Let $a$, $b$, and $m$ be positive integers such that $\gcd(a,b,m)=1$, $a\ne b$, and $a+b\ne m$. Let $\mathrm{cp}_{a,b,m}(n)$ denote the number of $(a,b,m)$-copartitions of $n$. Hal…

Let $a$ and $m$ be positive integers with $m$ odd and $\gcd(a,m)=1$, and let $\mathrm{cp}_{a,a,m}(n)$ denote the number of $(a,a,m)$-copartitions of $n$. Half-density conjecture. T…

Let $a$ and $m$ be positive integers, with $m$ even and $a$ odd, and let $\mathrm{cp}_{a,a,m}(n)$ denote the number of $(a,a,m)$-copartitions of $n$. Parity-density conjecture. … T…

The 2-parity conjecture. Assuming $E[2]\subset E(\mathbb{Q})$,

Ballantine–Merca conjecture. The equivalence

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.…

Let $n$ be the number of agents and $m$ the number of bads in a mixed manna division problem, and let $CU(\mathcal{P})$ denote the set of competitive utilities for a problem…