Das et al.'s congruence conjecture for generalized overcubic partitions

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Let aˉc(n)\bar a_c(n) denote the generalized overcubic partition function, with cc a positive integer. For all integers n≥0n\geq 0 and i≥1i\geq 1, the following congruences are conjectured:

aˉ3i+2(9n+2)≡0(mod6),aˉ3i+2(9n+5)≡0(mod6),aˉ3i+2(9n+8)≡0(mod6),aˉ9i+5(9n+3)≡0(mod12),aˉ9i+8(9n+3)≡0(mod12).\begin{aligned} \bar a_{3i+2}(9n+2)&\equiv 0 \pmod{6},\\ \bar a_{3i+2}(9n+5)&\equiv 0 \pmod{6},\\ \bar a_{3i+2}(9n+8)&\equiv 0 \pmod{6},\\ \bar a_{9i+5}(9n+3)&\equiv 0 \pmod{12},\\ \bar a_{9i+8}(9n+3)&\equiv 0 \pmod{12}. \end{aligned}

Das et al.'s conjecture. For all n≥0n\geq 0 and i≥1i\geq 1, these five congruences hold.

The paper states that it provides elementary proofs of the conjectures proposed by Das et al.; thus the conjecture is resolved by the source paper. The congruences are part of the study of generalized overcubic partitions and extend known congruence families for the function aˉc(n)\bar a_c(n).

References

Primary source

Suparno Ghoshal and Arijit Jana, “Combinatorial proof of a result on generalized overcubic partitions and related conjectures”, arXiv:2512.04775 (2025).

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