543 problems

Let $p_r(n)$ denote the coefficients in the partition-generating series used in the paper. For a positive integer $\lambda$, let $\mu_{\lambda}$ and $\delta_{\lambda}$ satisfy … Hi…

Determinant conjecture.

Further generalized cubic partition congruence conjecture.

Let $n\geq 0$, $i\geq 3$, and $k\geq 0$. The overcubic partition tuple function $\overline{b}_m(r)$ is considered at the indicated arithmetic progressions. Congruence conjecture. F…

Let $d_k(n)$ denote the $k$-elongated plane partition function, and let $c,n\geq 0$. For a list of integers in an argument, interpret the corresponding congruence as holding for ea…

Theta-lift nonvanishing conjecture. If $L(f,1)=0$, then

Hirschhorn–Sellers conjecture. For all $n\geq 1$ with $3\nmid n$,

Let $p_t(n)$ denote the number of $t$-multipartitions of $n$. Let $a$ and $t$ be odd integers, with $3\mid t$ whenever $3\mid a$. Define $b$ by … when $3\mid t$, and by…

Let … be an eta-quotient shifted so that all powers of $q$ are integral, and let $\delta_F$ be the odd density of its coefficients $c(n)$, when this density exists. A polynomial is…

Let $b_m(n)$ be the coefficients of the generating function for $m$-regular partitions. For even $m$, write … where $m_0$ is odd. The odd density of a sequence is the limiting prop…

Let $m_0$ be the odd part of an even integer $m$, and let $b_m(n)$ denote the coefficients of the corresponding $m$-regular partition generating function. Theorem gives congruences…

Let $n$, $k$, and $N$ be positive integers, let $r$ be an even integer, and let $p$ be a prime. Let $\chi$ be a quadratic Dirichlet character modulo $N$ satisfying…

Let $d\geq 3$ and let the theory be a unitary bosonic conformal field theory that is not a topological quantum field theory. Let $\mathcal{E}(\tau)$ denote its dimensionless modula…

Let $d\geq 3$ and let the theory be a unitary bosonic conformal field theory that is not a topological quantum field theory. Let $\mathcal{E}(\tau)$ denote its dimensionless modula…

Let $\overline{h}_k(x)=\nu_k\,p_{k,1}(x)\,p_{k,2}(x)\cdots p_{k,\alpha}(x)=\nu_k\,\widetilde{p}_k(x)$, where each $p_{k,n}$ is monic and $\nu_k$ is rational. Write the OEIS sequenc…

Let $A_{\mathcal K,k,m}(0)$ denote the constant term for the second family of meromorphic modular forms. K-family prime-parameter conjecture. The following assertions hold: if…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term, and let $d_3(k)$ be the sum of the base-3 digits of $k$. 3-adic valuation conjecture. If $m\equiv0\pmod3$,…

Let $d_2(k)$ be the sum of the binary digits of $k$. Two-digit valuation conjecture. If $d_2(k)=2$, $m=4j+2$ for $j=1,2,\ldots$, and $a=\operatorname{ord}_2(m)=1$, then … This is a…

Catalan valuation conjecture. Then $t=C_{1,n}$. The source presents this as an observed pattern in its empirical study of constant-term valuations; no resolution is given.

Let $d_2(k)$ be the sum of the binary digits of $k$, let $a=\operatorname{ord}_2(m)$, and let $o=\operatorname{ord}_2(A_{\overline{\mathcal K},k,m}(0))$. Binary-digit valuation con…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. Odd-prime valuation conjecture. If $p>2$ is prime and $a$ is a positive integer, then … This is an empiric…

Let $A_{\overline{\mathcal K},k,m}(0)$ denote the relevant constant term. 2-adic valuation conjecture at $k=2$. For every integer $a\geq2$, … This is an empirical special case of t…

Let $\mathcal{K}$ and $\overline{\mathcal{K}}$ be the two families of meromorphic modular forms in the source, and let $A_{\mathcal{K},k,m}(0)$ and…

Let $b_{3,e}(n)$ and $b_{3,o}(n)$ denote the numbers of $3$-regular partitions of $n$ having, respectively, an even and an odd number of parts. Let $n\geqslant 0$, let…

For every positive integer $n$, let $\operatorname{TMF}_0(n)$ denote the associated spectrum of topological modular forms and let $\operatorname{MF}_0(n)_0$ denote the degree-zero…