Integer partitions detect the primes.
basic_science · Level V
Where this comes from
- Record sourced from PubMed, PMID 39302965.
- Also identified by DOI 10.1073/pnas.2409417121 and PMC identifier 11441547.
- Licence recorded as CC BY-NC-ND.
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Abstract
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 <i>n</i> ≥ 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].