For each positive integer n, let O(n) be the set of overpartitions of n, and define operatornameOSOMEo(n)=∑_lambda∈O(n)∑_substackj is an overlined part of lambda j oddj and operatornameOSOMEe(n)=∑_lambda∈O(n)∑_substackj is an overlined part of lambda j evenj. The Baruah–Gogoi conjecture asserts congruences modulo 7 for the difference operatornameOSOMEo(n)-operatornameOSOMEe(n). The supplied source does not state the individual residue classes or congruence formulas.

Baruah–Gogoi conjecture on sums of odd and even overlined parts

For each positive integer $n$ , let $\mathcal{O}(n)$ be the set of overpartitions of $n$ , and define $\operatorname{OSOMEo}(n)=\sum_{\lambda\in\mathcal{O}(n)}\sum_{\substack{j\text{ is an overlined part of }\lambda\\ j\text{ odd}}}j$ and $\operatorname{OSOMEe}(n)=\sum_{\lambda\in\mathcal{O}(n)}\sum_{\substack{j\text{ is an overlined part of }\lambda\\ j\text{ even}}}j$ . The Baruah–Gogoi conjecture asserts congruences modulo $7$ for the difference $\operatorname{OSOMEo}(n)-\operatorname{OSOMEe}(n)$ . The supplied source does not state the individual residue classes or congruence formulas.

References

Primary source

arXiv

Additional references

A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts — arXiv — Eric H. Liu, X. L. Liu, Olivia X. M. Yao

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint is reported to prove the conjecture, but the result has not been independently checked.

The Baruah–Gogoi conjecture predicts divisibility congruences for the difference between odd and even overlined-part sums in overpartitions. The relevant statistic and conjectures appear in work by Nayandeep Deka Baruah and Pankaj Gogoi.

September 23, 2026 claimed proof

Eric H. Liu, X. L. Liu, and Olivia X. M. Yao are reported to prove the conjectured congruences using specializations of Watson’s quintuple product identity and related generating-function methods. An earlier September 18 preprint proves several congruences but labels its modulo- $7$ cases as numerical-evidence conjectures; the newer report purports to settle them. The proof has not undergone peer review.

Current status (as of September 2026): A complete proof is claimed by Liu, Liu, and Yao, but it remains unverified; independent confirmation and peer-reviewed validation are absent.

Sources

No solutions have been posted yet.