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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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS Vol. 121; no. 39; p. e2409417121
Main Authors: Craig, William, van Ittersum, Jan-Willem, Ono, Ken
Format: Journal Article
Language:English
Published: United States 24.09.2024
Subjects:
partitions
quasimodular forms
primes
ISSN:1091-6490, 1091-6490
Online Access:Get more information
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Summary: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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ISSN:1091-6490
1091-6490
DOI:10.1073/pnas.2409417121