Let overlinep_ell(n) denote the 2-color overpartition function, where ell is a positive integer, and let n≥0. The mod 8 congruence conjecture. For all ell equiv0,1 pmod4 and n≥0, overlinep_ell(4n+3)equiv0 pmod8. The case ell equiv0 pmod8 is established by the paper's Corollary 1, while the remaining cases are supported by numerical evidence and remain open.

The mod 8 congruence conjecture for the 2-color overpartition function

Less than 1 year old · traced to 2026

Let $\overline{p}_{\ell}(n)$ denote the $2$ -color overpartition function, where $\ell$ is a positive integer, and let $n\ge0$ . The mod 8 congruence conjecture. For all $\ell\equiv0,1\pmod4$ and $n\ge0$ ,

\overline{p}_{\ell}(4n+3)\equiv0\pmod8.

The case $\ell\equiv0\pmod8$ is established by the paper's Corollary 1, while the remaining cases are supported by numerical evidence and remain open.

References

Primary source

H. S. Sumanth Bharadwaj, N. Sujatha and S. Chandankumar, “Arithmetic properties of the 2-color overpartition function p_(n)”, arXiv:2607.16608 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 paper proves one subfamily, while a reader-written complete proof of the conjecture remains unverified.

The conjecture predicts that the $2$ -color overpartition function vanishes modulo $8$ at indices $4n+3$ for every $ll\equiv0,1\pmod4$ . Bharadwaj, Sujatha, and Chandankumar state it in their 2026 paper, which proves only a subfamily and leaves the rest open.

Known results

$ll\equiv0\pmod8$ : proved via Corollary $3.1$ ; $ll\equiv1\pmod4$ and $ll\equiv4\pmod8$ have numerical support but are not proved in the paper.

Posted attempt

A reader-written argument claims a complete proof by expanding the generating function modulo $8$ and observing that the relevant square-exponent sums avoid residue class $3\pmod4$ . This is an unverified complete-proof claim and is not independent evidence of resolution.

Current status (as of August 2026): The conjecture is proved for $ll\equiv0\pmod8$ , while the cases $ll\equiv1\pmod4$ and $ll\equiv4\pmod8$ remain mathematically open despite the unverified posted proof claim.

Sources

ProofThis solution needs a summarySee full solutionHide full solution

Let

f_d(q)=\prod_{j\ge1}(1-q^{dj}).

The defining generating function is

P_\ell(q) = \sum_{n\ge0}\overline p_\ell(n)q^n = \frac{f_2(q)f_{2\ell}(q)} {f_1(q)^2f_\ell(q)^2}.

Jacobi's identity gives

\varphi(-q)=\frac{f_1(q)^2}{f_2(q)},

and therefore

P_\ell(q) = \frac1{\varphi(-q)\varphi(-q^\ell)}.

Write

U(q)=\sum_{j\ge1}(-1)^jq^{j^2}, \qquad V(q)=U(q^\ell),

so that

\varphi(-q)=1+2U(q),\qquad \varphi(-q^\ell)=1+2V(q).

Expanding both inverses coefficientwise modulo $8$ yields

\begin{aligned} P_\ell(q) &=(1+2U)^{-1}(1+2V)^{-1}\\ &\equiv 1-2(U+V)+4(U^2+UV+V^2) \pmod8. \tag{1} \end{aligned}

Suppose

\ell\equiv0\ \text{or}\ 1\pmod4.

Every exponent in $U$ is a square and hence belongs to

\{0,1\}\pmod4.

Every exponent in $V$ is $\ell$ times a square, and therefore likewise belongs to $\{0,1\}\pmod4$ . Consequently the exponents in the linear terms $U,V$ belong to $\{0,1\}\pmod4$ , while those in the quadratic terms $U^2,UV,V^2$ belong to

\{0,1,2\}\pmod4.

None is congruent to $3\pmod4$ . Taking the coefficient of $q^{4n+3}$ in (1) proves

\boxed{\overline p_\ell(4n+3)\equiv0\pmod8}

for every $n\ge0$ and every positive $\ell\equiv0,1\pmod4$ , establishing the full conjecture.