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 M₀ (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.