Banerjee–Bringmann–Bachraoui's family of congruence conjectures for two-color partition numbers

From papers

Let c(n)c(n) be defined by

n=0c(n)qn=n=0(q2n+2;q2)(q2n+1;q2)2q2n+1.\sum_{n=0}^\infty c(n)q^n=\sum_{n=0}^\infty \frac{(-q^{2n+2};q^2)_\infty}{(q^{2n+1};q^2)_\infty^2}q^{2n+1}.

Banerjee–Bringmann–Bachraoui's conjecture. For all integers nn and kk,

c(22k+3n+114k+13)0(mod4),c\left(2^{2k+3}n+\frac{11\cdot 4^{k}+1}{3}\right)\equiv 0\pmod 4, c(22k+3n+174k+13)0(mod8),c\left(2^{2k+3}n+\frac{17\cdot 4^{k}+1}{3}\right)\equiv 0\pmod 8,

and

c(22k+4n+384k+13)0(mod4).c\left(2^{2k+4}n+\frac{38\cdot 4^{k}+1}{3}\right)\equiv 0\pmod 4.

These congruences extend the proposed infinite families of congruences for the limiting two-color partition function. The supplied context does not state that this family has been resolved.

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Sources & referencesView supporting material

Primary source

Junjie Sun and Olivia X. M. Yao, “Proof of a conjecture of Banerjee,Bringmann and Bachraoui on infinite families of congruences”, arXiv:2604.05403 (2026).

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