Let c(n) denote the coefficient sequence under consideration, and let n∈N_0. The congruence conjecture for c(32n+23). c(32n+23)equiv 0 ( mod , 8 ). This conjecture is proposed as an open question for future research. The authors report numerical verification for 32n+23≤ 5000, but no proof is supplied in the provided text.

Congruence conjecture for c(32n+23) modulo 8

From papers

Let $c(n)$ denote the coefficient sequence under consideration, and let $n\in\mathbb{N}_0$ . The congruence conjecture for $c(32n+23)$ .

c(32n+23)\equiv 0\ \left( \mathrm{mod} \, 8 \right).

This conjecture is proposed as an open question for future research. The authors report numerical verification for $32n+23\le 5000$ , but no proof is supplied in the provided text.

Progress summary

Solved

A 2026 preprint claims to prove the conjecture, but the proof has not yet been independently verified.

Banerjee, Bringmann, and El Bachraoui proposed that $c(32n+23)\equiv0\pmod{8}$ for every $n\in\mathbb{N}_0$ . Their April 2026 paper reports numerical verification through $32n+23\le5000$ but labels the assertion Conjecture 8.1 and supplies no proof.

2026 preprint claiming a proof

Juejie Sun and Olivia X. M. Yao claim to prove a broader conjecture of Banerjee, Bringmann, and El Bachraoui; their stated infinite family includes the present congruence as the case $k=1$ . This is a claimed preprint proof, not an independently corroborated resolution in the retrieved material.

Current status (as of August 2026): The congruence is numerically verified and is covered by Sun–Yao's claimed 2026 preprint proof, but independent verification remains outstanding.

Sources

Sources & referencesView supporting material

Primary source

Koustav Banerjee, Kathrin Bringmann and Mohamed El Bachraoui, “On congruence conjectures of Andrews and Bachraoui”, arXiv:2604.02239 (2026).

1 week ago

Prior proof and attribution. This conjecture was already proved by Sun and Yao in arXiv:2604.05403, Lemma 2.2, equation (2.22). The original problem is therefore not open. The proof below is only an alternative derivation through a stronger residue-three transfer identity; it does not claim the first proof.

The congruence follows from the stronger transfer identity

\boxed{ c(4m+3)\equiv-p_\omega(4m+3)\pmod8 \qquad(m\ge0), }

where the distinct mock-theta partition function $p_\omega$ is defined by

q\omega(q)=\sum_{N\ge1}p_\omega(N)q^N.

The primary source gives

C(q)=\sum_{N\ge0}c(N)q^N = 2qA(q)B(-q)-q\omega(-q),

and its equation (5.5) implies

A(q)\equiv A_0(q^4)+qA_1(q^4)\pmod4, \qquad A_1(q)\in2\mathbb Z[[q]].

Write $B(q)=\sum_{j=0}^3q^jB_j(q^4)$ . The exact published four-dissections of the same function $B$ are

B_1(q)=\frac{2f_2^8}{f_1^7}, \qquad B_2(q)=\frac{4f_2^2f_4^4}{f_1^5}, \qquad f_j=(q^j;q^j)_\infty.

Hence $B_1\in2\mathbb Z[[q]]$ and $B_2\in4\mathbb Z[[q]]$ . The residue-two part of $A(q)B(-q)$ is therefore

A_0B_2-A_1B_1\equiv0\pmod4.

The first term $2qA(q)B(-q)$ thus vanishes modulo $8$ on every exponent $4m+3$ , and the remaining term contributes

-[q^{4m+2}]\omega(-q) = -p_\omega(4m+3).

This proves the transfer identity.

Theorem 1.1 of Andrews–Passary–Sellers–Yee gives

p_\omega\left( 2^{2k+3}n+\frac{17\cdot4^k+1}{3} \right)\equiv0\pmod8

for all $k,n\ge0$ . Set $k=1$ , obtaining $p_\omega(32n+23)\equiv0\pmod8$ . Since $32n+23\equiv3\pmod4$ , the transfer identity yields

\boxed{c(32n+23)\equiv0\pmod8\qquad(n\ge0).}

Sources: Banerjee–Bringmann–El Bachraoui, arXiv:2604.02239, equations (4.1), (5.5), Conjecture 8.1; Andrews–Passary–Sellers–Yee, The Ramanujan Journal 43 (2017), 347–357, Theorem 1.1, doi:10.1007/s11139-016-9812-2; “On second order mock theta function $B(q)$ ,” Electronic Research Archive 30 (2022), 52–65, equations (1.6)–(1.7), doi:10.3934/era.2022003.

0 endorsements

Shivam Patel · 20 hours ago · edited