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Ramanujan-type congruences for the smallest parts function
Abstract
In a famous paper of Andrews, he defined and studied the smallest parts function spt(n). He showed that it satisfies the three Ramanujan-type congruences: spt(5n+4)≡0pmod5, spt(7n+5)≡0pmod7, spt(13n+6)≡0pmod13. In a recent preprint, the author showed that these are the only "canonical'' congruences. Here, we complete the classification of Ramanujan-type congruences for spt, showing that if spt(ℓ n+b)≡0pmodℓ for all n≥0, where ℓ≥5 is prime, then (ℓ,b)∈{(5,4),(7,5),(13,6)}. That is, Andrews' congruences are the only Ramanujan-type congruences for spt(n). These results have been verified in Lean by AxiomProver.
Related
- Class polynomial obstructions to canonical congruences for the smallest parts function
- The smallest-parts function and elliptic curves
- Congruences for Overcolored Partition k-tuples Restricted by Parity of the Parts
- Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
- Internal congruences modulo powers of 2 for k-colored overpartitions into odd parts
BibTeX
@misc{rolen2026ramanujantype,
title = {{Ramanujan-type congruences for the smallest parts function}},
author = {Larry Rolen},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04178},
eprint = {2611.04178},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04178},
}
RIS
TY - UNPB
TI - Ramanujan-type congruences for the smallest parts function
AU - Larry Rolen
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04178
UR - https://arxiv.org/abs/2611.04178
AB - In a famous paper of Andrews, he defined and studied the smallest parts function spt(n). He showed that it satisfies the three Ramanujan-type congruences: spt(5n+4)≡0pmod5, spt(7n+5)≡0pmod7, spt(13n+6)≡0pmod13. In a recent preprint, the author showed that these are the only "canonical'' congruences. Here, we complete the classification of Ramanujan-type congruences for spt, showing that if spt(ℓ n+b)≡0pmodℓ for all n≥0, where ℓ≥5 is prime, then (ℓ,b)∈{(5,4),(7,5),(13,6)}. That is, Andrews' congruences are the only Ramanujan-type congruences for spt(n). These results have been verified in Lean by AxiomProver.
ER -