Abstract: Integer partitions, the fundamental building blocks in additive number theory, detect prime numbers in an unexpected way. We explain 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) − 960 M_3(n) = 0, where the M_a(n) are MacMahon’s well-studied partition functions. Further, in order to explain how such equations arise we give a short introduction to quasi-shuffle algebras as well as modular forms and their associated Galois representations.