Explicit linear dependence congruence relations for the partition function modulo 4
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915076758306816 |
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| author | Charlton, Steven |
| author_facet | Charlton, Steven |
| contents | 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 $k = 316, 317, 319, 321, 322, 326, \ldots$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_17459 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit linear dependence congruence relations for the partition function modulo 4 Charlton, Steven Number Theory Primary: 11P83. Secondary: 05A17 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 $k = 316, 317, 319, 321, 322, 326, \ldots$. |
| title | Explicit linear dependence congruence relations for the partition function modulo 4 |
| topic | Number Theory Primary: 11P83. Secondary: 05A17 |
| url | https://arxiv.org/abs/2412.17459 |