2024/12/23 by Steven J. Charlton, Charlton, Steven
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Primary: 11P83. Secondary: 05A17
paper · pdf · doi:10.48550/arxiv.2412.17459
openalex publication_date 2024/12/23 · openalex created_date 2024/12/25 · openalex updated_date 2026/07/28
Almost nothing is known about the parity of the partition function p(n), which is conjectured to be random. Despite this expectation, Ono 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 ≤ 24k-1 for k=309 (resp. k=312); new relations occur for k = 316, 317, 319, 321, 322, 326, ….