AN APPLICATION OF SCHLÀFLPS MODULAR EQUATION TO A CONJECTURE OF RAMANUJANf

In 1918 RamanujanJ made the following conjecture: If ç = 5, 7, or 11, and if 24^ — 1 is divisible by q, then the number p(n) of unrestricted partitions of n is divisible by q. Ramanujan himself proved this conjecture to be true in casej q = S, 7, 5, and 7, and also§ for q=ll and l l 2 . It has since been proved|| for q = 5. Some modification of the conjecture is necessary, however, since, as Chowla^f was first to notice, it fails for q = 7. In fact, since 24 • 243 — 1 = 5831 is divisible by 7, it would follow from the conjecture that ^(243) is also divisible by 7. However, Gupta's table** of p(n) gives