Integer partitions detect the primes

TL;DR

This paper reveals that certain partition functions in additive number theory can uniquely identify prime numbers through specific equations, establishing a novel link between partitions and primality detection.

Contribution

It demonstrates that prime numbers can be characterized by solutions to equations involving partition functions, introducing a new method to detect primes using additive number theory.

Findings

Primes are solutions to special equations involving partition functions.

Infinite families of prime-detecting equations exist for MacMahonesque partition functions.

Partition functions can serve as tools for prime detection in number theory.

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 $n\geq 2$ 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 $M_{\vec{a}}(n),$ we prove that there are infinitely many such prime detecting equations with constant coefficients, such as $$ 80M_{(1,1,1)}(n)-12M_{(2,0,1)}(n)+12M_{(2,1,0)}(n)+\dots-12M_{(1,3)}(n)-39M_{(3,1)}(n)=0. $$