Let c(n) denote the coefficient sequence under consideration, and let k,n∈N_0. The infinite-family congruence conjecture for c(n). c(2^2k+3n+frac11· 4^k+13)equiv 0 ( mod , 4 ), c(2^2k+3n+frac17· 4^k+13)equiv 0 ( mod , 8 ), c(2^2k+4n+frac38· 4^k+13)equiv 0 ( mod , 4 ). The authors propose these congruences as a more general infinite family related to the preceding open congruence for c(n). The provided text does not state a proof or resolution.

Infinite family of congruence conjectures for c(n)

Less than 1 year old · traced to 2026

Let $c(n)$ denote the coefficient sequence under consideration, and let $k,n\in\mathbb{N}_0$ . The infinite-family congruence conjecture for $c(n)$ .

c\left(2^{2k+3}n+\frac{11\cdot 4^k+1}{3}\right)\equiv 0\ \left( \mathrm{mod} \, 4 \right),

c\left(2^{2k+3}n+\frac{17\cdot 4^k+1}{3}\right)\equiv 0\ \left( \mathrm{mod} \, 8 \right),

c\left(2^{2k+4}n+\frac{38\cdot 4^k+1}{3}\right)\equiv 0\ \left( \mathrm{mod} \, 4 \right).

The authors propose these congruences as a more general infinite family related to the preceding open congruence for $c(n)$ . The provided text does not state a proof or resolution.

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 all three infinite congruence families, while a separate posted alternative proof remains independently unverified.

Banerjee, Bringmann, and El Bachraoui proposed the three infinite families in 2026 after establishing the initial cases. The conjecture asserts divisibility patterns for $c(n)$ across every $k,n\ge0$ .

April 2026 claimed proof

Sun and Yao claim a complete proof for all three families, using results from Banerjee, Bringmann, and El Bachraoui together with a Watson identity. Their preprint also recovers the established cases $c(8n+4)\equiv0\pmod4$ , $c(8n+6)\equiv0\pmod8$ , and $c(16n+13)\equiv0\pmod4$ . No independent verification, objection, or retraction was found.

Posted attempt

A posted alternative derivation claims a complete proof via a transfer identity relating $c(n)$ to a mock-theta coefficient sequence and previously known congruences. The attempt has not been independently verified.

Current status (as of August 2026): The initial cases are established, while Sun and Yao claim the full three-family conjecture is proved; independent verification is not recorded.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

Prior proof and attribution. All three conjectured families were already proved by Sun and Yao in arXiv:2604.05403, Section 2. 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.

All three congruence families hold for every $k,n\ge0$ . The key is a transfer theorem between the source's new coefficient sequence $c$ and the distinct mock-theta partition function $p_\omega$ :

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

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

Indeed, the primary source gives

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

and

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

For $B(q)=\sum_{j=0}^3q^jB_j(q^4)$ , the exact classical four-dissections 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.

Thus $B_1$ is even and $B_2$ is divisible by $4$ . Consequently the residue-two part of $A(q)B(-q)$ satisfies

A_0B_2-A_1B_1\equiv0\pmod4.

Therefore $2qA(q)B(-q)$ contributes zero modulo $8$ to every exponent $4m+3$ , while

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

proving the transfer theorem.

Now define

\begin{aligned} N_1(k,n) &= 2^{2k+3}n+\frac{11\cdot4^k+1}{3},\\ N_2(k,n) &= 2^{2k+3}n+\frac{17\cdot4^k+1}{3},\\ N_3(k,n) &= 2^{2k+4}n+\frac{38\cdot4^k+1}{3}. \end{aligned}

For every $k\ge1$ , all three integers satisfy $N_i(k,n)\equiv3\pmod4$ . The published theorem of Andrews–Passary–Sellers–Yee gives

p_\omega(N_1(k,n))\equiv0\pmod4,

p_\omega(N_2(k,n))\equiv0\pmod8,

and

p_\omega(N_3(k,n))\equiv0\pmod4.

Applying the transfer identity proves the corresponding three congruences for $c$ for every $k\ge1$ .

When $k=0$ , the three statements reduce to

c(8n+4)\equiv0\pmod4, \qquad c(8n+6)\equiv0\pmod8, \qquad c(16n+13)\equiv0\pmod4,

which are already established as Theorems 1.4–1.6 in the primary source. Hence all three conjectured families hold for every $k,n\ge0$ .

The 2017 theorem concerns $p_\omega$ , not $c$ ; Sun and Yao already proved the stated conjecture in arXiv:2604.05403, and the transfer identity gives an alternative stronger route.

Sources: Banerjee–Bringmann–El Bachraoui, arXiv:2604.02239, Conjecture 8.2; 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.