Ramanujan's congruences and Dyson's crank.

T he achievement of Karl Mahlburg in this issue of PNAS (1) adds a lustrous chapter to a unique mathematical object: the crank. In 1944, the crank was first hinted at by Freeman Dyson (2), then an undergraduate at Cambridge University. He had written an article, titled Some Guesses in the Theory of Partitions, for Eureka, the undergraduate mathematics journal of Cambridge. Dyson discovered the many conjectures in this article by attempting to find a combinatorial explanation of Ramanujan’s famous congruences for p(n), the number of partitions of n. The three simplest of Ramanujan’s congruences assert that: