In 1742, firstly Leonhard Euler invented the generating function for P(n), where P(n) is the number of partitions of n [P(n) is defined to be 1]. Srinivasa Ramanujan was born on 22 December 1887. In 1916, S. Ramanujan invented the generating function for P(n) (2nd time). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). MacMahon established a table of P(n) for the first 200 values of n, and Ramanujan observed that the table indicated certain simple congruences properties of P(n). 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.
Congruences, Enumerating, Modulo, Residues, Ramanujan’s Lost Notebook
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