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

From papers

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.

Progress summary

Open

The conjecture remains open: one family of cases is proved, while the others have only numerical support.

The conjecture predicts that the relevant counting function vanishes modulo $8$ at every index of the form $4n+3$ whenever $\ell$ is congruent to $0$ or $1$ modulo $4$ . The recent literature records a proof only for the subfamily $\ell\equiv0\pmod8$ ; the remaining cases are explicitly left open.

Known results

$\ell\equiv0\pmod8$ : proved in the cited paper via a corollary; the cases $\ell\equiv1\pmod4$ and $\ell\equiv4\pmod8$ remain supported only by computation.

July 2026 paper

H. S. Sumanth Bharadwaj, N. Sujatha, and S. Chandankumar state the full conjecture in their paper on arithmetic properties of the $2$ -color overpartition function, but report no proof beyond $\ell\equiv0\pmod8$ and describe the remaining cases as numerical evidence. No counterexample, claimed proof, or verification was found.

Current status (as of August 2026): The congruence is proved for $\ell\equiv0\pmod8$ , while the cases $\ell\equiv1\pmod4$ and $\ell\equiv4\pmod8$ remain open.

Sources

Sources & referencesView supporting material

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).

1 week ago

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.

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Shivam Patel · 22 hours ago