PreprintCC BY 4.0arXiv
Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences
Abstract
In 2015, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t̅(n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In their work, they proved that t̅(n) satisfies an elegant congruence modulo 3. Since then, a number of authors have studied arithmetic properties satisfied by t̅(n). In particular, in 2023, Hanson and Smith utilized the theory of modular forms to prove the following two congruences modulo 5: For all n≥ 0, t̅(80n+40)≡ t̅(80n+60)≡ 0 pmod5. Our goal in this work is to provide a truly elementary proof of this pair of congruences.
Related
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BibTeX
@misc{paudel2026elementary,
title = {{Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences}},
author = {Bishnu Paudel and James A. Sellers and Haiyang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16376},
eprint = {2609.16376},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16376},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elementary proofs of congruences modulo 5 for overpartitions with restricted odd differences
AU - Bishnu Paudel
AU - James A. Sellers
AU - Haiyang Wang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16376
UR - https://arxiv.org/abs/2609.16376
AB - In 2015, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function t̅(n) to be the number of overpartitions of weight n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In their work, they proved that t̅(n) satisfies an elegant congruence modulo 3. Since then, a number of authors have studied arithmetic properties satisfied by t̅(n). In particular, in 2023, Hanson and Smith utilized the theory of modular forms to prove the following two congruences modulo 5: For all n≥ 0, t̅(80n+40)≡ t̅(80n+60)≡ 0 pmod5. Our goal in this work is to provide a truly elementary proof of this pair of congruences.
ER -