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Nonexistence of Constant Congruences for the Cumulative Partition Function
Abstract
Let p(n) be the ordinary partition function and let c(n)=∑k=0n−1p(k), with c(0)=0. We prove that, for every a≥1, b≥0, and m≥2, the sequence c(an+b) is not eventually constant modulo m, thereby answering a question posed by Chen. The proof combines the algebraicity over ℂ(j) of modular functions for finite-index subgroups of SL2(ℤ) with Voloch's theorem on the transcendence of elliptic modular functions in positive characteristic.
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BibTeX
@misc{ji2026nonexistence,
title = {{Nonexistence of Constant Congruences for the Cumulative Partition Function}},
author = {Kathy Q. Ji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31102},
eprint = {2610.31102},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31102},
}
RIS
TY - UNPB
TI - Nonexistence of Constant Congruences for the Cumulative Partition Function
AU - Kathy Q. Ji
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31102
UR - https://arxiv.org/abs/2610.31102
AB - Let p(n) be the ordinary partition function and let c(n)=∑_k=0^n−1p(k), with c(0)=0. We prove that, for every a≥1, b≥0, and m≥2, the sequence c(an+b) is not eventually constant modulo m, thereby answering a question posed by Chen. The proof combines the algebraicity over ℂ(j) of modular functions for finite-index subgroups of SL_2(ℤ) with Voloch's theorem on the transcendence of elliptic modular functions in positive characteristic.
ER -