A quadratic-residue dichotomy for two partition functions modulo 3
Abstract
Let f0,1,4(n) denote the number of partitions of n into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let f0,2,3(n) be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime p ≡ 3 pmod 4 there are explicit non-negative integers a(p) and b(p), determined by 20a(p) ≡ −9 and 20b(p) ≡ −1 pmodp2, such that f0,1,4(p2m + a(p)) and f0,2,3(p2m + b(p)) are congruent modulo 3 to f0,1,4(m) and f0,2,3(m) when p ≡ ± 1 pmod 5, and to f0,2,3(m) and f0,1,4(m) when p ≡ ± 2 pmod 5. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo p, and interchanged otherwise. The smallest cases are f0,1,4(9m) ≡ f0,2,3(m) and f0,2,3(9m+4) ≡ f0,1,4(m). The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of (1−x)ℓ ≡ 1−xℓ pmodℓ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that −1 is a quadratic non-residue modulo p. As corollaries we obtain, for each such p, a congruence with exceptions on a progression of modulus p, and an infinite family of self-similarity congruences on progressions of modulus pdn with d = 2 or d = 4 and constants 9(pdn−1)/20 and (pdn−1)/20.
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BibTeX
@misc{li2026quadraticresidue,
title = {{A quadratic-residue dichotomy for two partition functions modulo 3}},
author = {Jiyuan Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22340},
eprint = {2609.22340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22340},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A quadratic-residue dichotomy for two partition functions modulo 3
AU - Jiyuan Li
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.22340
UR - https://arxiv.org/abs/2609.22340
AB - Let f_0,1,4(n) denote the number of partitions of n into parts congruent to 0, 1 or 4 modulo 5 with each part used at most twice, and let f_0,2,3(n) be defined analogously for parts congruent to 0, 2 or 3 modulo 5. We prove that for every prime p ≡ 3 pmod 4 there are explicit non-negative integers a(p) and b(p), determined by 20a(p) ≡ −9 and 20b(p) ≡ −1 pmodp^2, such that f_0,1,4(p^2m + a(p)) and f_0,2,3(p^2m + b(p)) are congruent modulo 3 to f_0,1,4(m) and f_0,2,3(m) when p ≡ ± 1 pmod 5, and to f_0,2,3(m) and f_0,1,4(m) when p ≡ ± 2 pmod 5. By quadratic reciprocity the two functions are preserved exactly when 5 is a quadratic residue modulo p, and interchanged otherwise. The smallest cases are f_0,1,4(9m) ≡ f_0,2,3(m) and f_0,2,3(9m+4) ≡ f_0,1,4(m). The proof reduces each generating function modulo 3 to the square of a Rogers-Ramanujan-type theta function by means of (1−x)^ℓ ≡ 1−x^ℓ pmodℓ and the Jacobi triple product, and then dissects the resulting binary quadratic form using the fact that −1 is a quadratic non-residue modulo p. As corollaries we obtain, for each such p, a congruence with exceptions on a progression of modulus p, and an infinite family of self-similarity congruences on progressions of modulus p^dn with d = 2 or d = 4 and constants 9(p^dn−1)/20 and (p^dn−1)/20.
ER -