Year: 2025
Type: article
Abstract: Almost nothing is known about the parity of the partition function p(n), which is conjectured to be random. Despite this expectation, Ono [10] surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4 for p(n), indicating that the parity of the partition function cannot be truly random. Answering a question of Ono, we explicitly exhibit the first examples of these relations which he proved theoretically exist. The first two relations invoke 131 (resp. 198) different discriminants $$ D \le 24 k - 1 $$ D ≤ 24 k - 1 for $$ k = 309 $$ k = 309 (resp. $$ k = 312 $$ k = 312 ); new relations occur for $$ k = 316,\, 317,\, 319,\, 321,\, 322,\, 326, \, \ldots $$ k = 316 , 317 , 319 , 321 , 322 , 326 , … .
Source: Research in Number Theory
Author: Steven Charlton
Institution: Max Planck Institute for Mathematics
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,590
Funder: Universität zu Köln
OpenAlex ID: W4407821965