A note on congruences for generalized cubic partitions modulo primes
Russelle Guadalupe
TL;DR
The paper studies congruences for the generalized cubic partition function $a_c(n)$, extending beyond the known modular-form proofs of $a_3(7n+4)\equiv 0\pmod{7}$ and $a_5(11n+10)\equiv 0\pmod{11}$ by providing a purely $q$-series derivation. It employs classical identities from Euler and Ramanujan, together with modular reductions $f_{2p}\equiv f_2^p\pmod{p}$, to recover the original congruences and to extract new infinite families of congruences for primes $p\not\equiv 1\pmod{8}$. The first new family covers primes $p\equiv 5,7\pmod{8}$ with $p\mid 8l+3$, giving $a_{p-4}(pn+l)\equiv 0\pmod{p}$, while the second uses an Ahlgren-type result to obtain $a_{p-6}\big(pn+\frac{13(p^2-1)}{24}\big)\equiv 0\pmod{p}$ for primes $p\equiv 3,7\pmod{8}$. Together, these results broaden the landscape of congruences for generalized cubic partitions via elementary $q$-series methods. This work enhances the toolkit for studying partition congruences by linking Ramanujan-type identities to modular-arithmetic extraction without invoking modular-form machinery.
Abstract
Recently, Amdeberhan, Sellers, and Singh introduced the notion of a generalized cubic partition function $a_c(n)$ and proved two isolated congruences via modular forms, namely, $a_3(7n+4)\equiv 0\pmod{7}$ and $a_5(11n+10)\equiv 0\pmod{11}$. In this paper, we provide another proof of these congruences by using classical $q$-series manipulations. We also give infinite families of congruences for $a_c(n)$ for primes $p\not\equiv 1\pmod{8}$.