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arXiv · 2412.17459

Explicit linear dependence congruence relations for the partition function modulo 4

Steven Charlton
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Abstract

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 $D \leq 24k-1$ for $k=309$ (resp. $k=312$); new relations occur for $k = 316, 317, 319, 321, 322, 326, \ldots$.

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Steven Charlton. 2024-12-23. Explicit linear dependence congruence relations for the partition function modulo 4. https://arxiv.org/abs/2412.17459

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