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

Less than 1 year old · traced to 2026

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.

References

Primary source

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

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims to prove the pattern, and an unverified posted alternative derivation repeats that claim, but independent confirmation is absent.

Banerjee, Bringmann, and El Bachraoui proposed that the coefficient sequence satisfies the stated divisibility for every nonnegative index, recording numerical checks through $32n+23\le5000$ . Their paper’s abstract says the conjectures are settled, but its body labels this assertion Conjecture 8.1 and supplies no proof.

Known results

$c(8n+4)\equiv0\pmod4$ (Banerjee, Bringmann, and El Bachraoui, 2026).
$c(8n+6)\equiv0\pmod8$ (Banerjee, Bringmann, and El Bachraoui, 2026).
$c(16n+13)\equiv0\pmod4$ (Banerjee, Bringmann, and El Bachraoui, 2026).

April 2026 claimed proof. Sun and Yao’s preprint states the target congruence as equation (2.22), deriving it from a broader family; this is a claimed proof, not an independently verified result.

Posted attempt

An alternative derivation claims a stronger transfer identity and combines it with an earlier $p_\omega$ congruence to obtain the target statement. The attempt is not independently verified.

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

Sources

ProofThis solution needs a summarySee full solutionHide full solution

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.