Equation Evaluation
Partitions of 2
Prime Detection Across the Number Line
Equation Value vs. n — Primes hit zero
What is Ma(n)?
MacMahon's partition function Ma(n) sums the products of part multiplicities across all partitions of n that use exactly a distinct part sizes. For example, 6 = 2+2+1+1 has 2 distinct parts (sizes 1 and 2) with multiplicities 2 and 2, contributing 2×2 = 4 to M₂(6).
Why is this surprising?
Primes are defined by multiplication (not divisible by smaller numbers), yet they are precisely detected by partition functions — objects from additive number theory. This bridges the multiplicative and additive worlds of mathematics in an unexpected way.
The Discovery
Craig, van Ittersum, and Ono proved that infinitely many such prime-detecting equations exist using MacMahon partition functions. The simplest is shown above. A second equation uses M₃(n): (3n³−13n²+18n−8)M₁(n) + (12n²−120n+212)M₂(n) − 960·M₃(n) = 0.
Historical Roots
The proof connects MacMahon's Partition Analysis (1916), the Hardy-Ramanujan asymptotic formula (1918), and the theory of quasimodular forms — tools so classical that Ono noted this result "could have been found in the 1950s."