Partition Congruences and the Andrews-Garvan-Dyson Crank
Ramanujan's Celebrated Congruences
~ A topic that originated about 100 years ago ~
Karl Mahlburg, a graduate student at the University of Wisconsin, has announced the proof of a conjecture by Ono. This work essentially asserts the "crank" of Andrews-Garvan-Dyson satisfies the same types of general congruences as Ramanujan's work on partition functions.
p(5n + 4) ≡ 0 (mod 5)
p(7n + 5) ≡ 0 (mod 7)
p(11n + 6) ≡ 0 (mod 11)
See Karl Mahlburg's "Partition congruences and the Andrew-Garvan-Dyson crank," PNAS, October 25, 2005, vol. 102, no. 43, 15373-15376.