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 <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].
Authors
- William Craig (ORCID: https://orcid.org/0000-0003-1808-6503)
- Jan‐Willem van Ittersum (ORCID: https://orcid.org/0000-0003-1541-0232)
- Ken Ono (ORCID: https://orcid.org/0000-0003-3670-319X)
Institutions
- TH Köln - University of Applied Sciences (DE)
- University of Cologne (DE)
- United States Naval Academy (US)
- University of Virginia (US)
Publication Details
- Journal
- Proceedings of the National Academy of Sciences
- Published
- 2024-09-20
- DOI
- https://doi.org/10.1073/pnas.2409417121
- Citations
- 8
- Primary Topic
- Advanced Mathematical Identities
- Type
- article
- Field-Weighted Citation Impact
- 10.16
Funders
- National Science Foundation
- European Commission
- Deutsche Forschungsgemeinschaft
- Universität zu Köln
- European Research Council
- HORIZON EUROPE European Research Council