Ramanujan’s Famous Partition Congruences

Abstract

In 1742, Firstly Leonhard Euler invented the generating function for , where  is the number of partitions of n [  is defined to be 1]. Srinivasa Ramanujan was born on 22 December 1887. In 1916, S. Ramanujan invented the generating function for  (2^nd time). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of . MacMahon established a table of  for the first 200 values of n, and Ramanujan observed that the table indicated certain simple congruences properties of . In 1916, S. Ramanujan quoted his famous partition congrucnecs. In particular, the numbers of the partitions of numbers 5m+ 4, 7m+5, and 11m +6 are divisible by 5, 7, and 11 respectively. Now this paper shows how to prove the Ramanujan’s famous partitions congruences modulo 5, 7, and 11 respectively.