Integer partitions detect the primes
We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer ≥ 2 is prime if and...
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| Vydáno v: | Proceedings of the National Academy of Sciences - PNAS Ročník 121; číslo 39; s. e2409417121 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
United States
24.09.2024
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| Témata: | |
| ISSN: | 1091-6490, 1091-6490 |
| On-line přístup: | Zjistit podrobnosti o přístupu |
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| Shrnutí: | We show that integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. Answering a question of Schneider, we show that the primes are the solutions to special equations in partition functions. For example, an integer
≥ 2 is prime if and only if [Formula: see text]where the [Formula: see text] are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions [Formula: see text] we prove that there are infinitely many such prime detecting equations with constant coefficients, such as [Formula: see text]. |
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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1091-6490 1091-6490 |
| DOI: | 10.1073/pnas.2409417121 |