Explicit linear dependence congruence relations for the partition function modulo 4

TL;DR

This paper explicitly constructs the first known examples of linear dependence relations modulo 4 for the partition function, challenging the assumption of its randomness and building on Ono's theoretical results.

Contribution

It provides explicit examples of linear dependence relations modulo 4 for the partition function, confirming Ono's existence results with concrete instances.

Findings

Explicit examples of linear dependence relations modulo 4 for p(n)

Relations involve discriminants up to 24k-1 for specific k values

New relations are identified for various k values beyond initial examples

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 $k = 316, 317, 319, 321, 322, 326, \ldots$.