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 n≥ 2 is prime if and only if (3n³ - 13n² + 18n - 8)M₁(n) + (12n² - 120n + 212)M₂(n) -960M₃(n) = 0, where the Mₐ(n) are MacMahon's well-studied partition functions. More generally, for "MacMahonesque" partition functions Ma⃗(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)+-12M(1,3)(n)-39M(3,1)(n)=0.