Modulo-16 congruences for 18-colored generalized Frobenius partitions

Establish, for every integer n≥0, the congruences cφ₁₈(30n+19)≡0 (mod 16) and cφ₁₈(30n+25)≡0 (mod 16).

Background

The paper studies congruence properties of the number cφ₁₈(n) of 18-colored generalized Frobenius partitions. Its principal proved result establishes cφ₁₈(3n+2)≡0 (mod 2187), along with additional congruences modulo 8 and 81. In the concluding remarks, the authors identify two further modulo-16 congruences suggested by numerical evidence.

These assertions are explicitly presented as conjectures, and the authors state that their proofs are left for future investigation. They therefore remain unresolved within the paper.

References

Based on numerical evidence, we conjecture the following congruences and leave their proofs for future investigation. For all $n\ge0$, \begin{align*} c\phi_{18}\left(30n+19\right)&\equiv 0\pmod{16},\ c\phi_{18}\left(30n+25\right)&\equiv 0\pmod{16}. \end{align*}

— Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions  (2609.05302 - Das et al., 4 Sep 2026) in Section "Concluding Remarks," item 1