Proof of a conjecture of Ramanujan

We write and so that p ( n ) is the number of unrestricted partitions of n . Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24 m ≡ 1 (mod q n ), then p( m ) ≡ 0 (mod q n ). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n , and a suitably modified form for q = 7 and all n . (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically available for the case q = 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ 0 ( q ) has genus zero, is applicable in principle to all primes q without prohibitive calculation. In particular he proved the conjecture for q = 11 and n = 3 in [7]. Here I shall prove the conjecture for q = 11 and all n , following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result for c ( m ), the Fourier coefficient‡ of Klein's modular invariant j (τ) as