Integer partitions detect the primes

doi:10.1073/pnas.2409417121

Integer partitions detect the primes

2024-09-20

1

OA

AI

W

William Craig

J

Jan‐Willem van Ittersum

K

Ken Ono *

DOI：10.1073/pnas.2409417121

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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 n >= 2 is prime if and only if (3n(3) - 13n(2) + 18n 8)M-1 (n)+(12n(2) 120n + 212)M-2 (n) 960M(3) (n) = 0, where the M-a(n) are MacMahon's well-studied partition functions. More generally, for MacMahonesque partition functions M-(a) over right arrow (n), we prove that there are infinitely many such prime detecting equations with constant coefficients, such as 80M((1,1,1))(n) 12M((2,0,1))(n) + 12M (2,1,0)(n) + center dot center dot center dot -12M((1,3))(n) 39M((3,1))(n) = 0.

Keywords:

partitions

primes

quasimodular forms

Integer partitions detect the primes

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