Ramanujan’s Famous Partition Congruences-Reference-Cited by-同舟云学术

Ramanujan’s Famous Partition Congruences

Published:2016-12-23 Issue:1 Volume:5 Page:145-158
ISSN:2307-9584
Container-title:Asian Journal of Applied Science and Engineering
language:
Short-container-title:Asian j. appl. sci. eng.

Author:

Hossain Md. Fazlee,Bhattacharjee Nil Ratan,Das Sabuj

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 (2nd 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.

Publisher

ABC Journals

Subject

Media Technology

Reference6 articles.

1. B.C. Berndt, Ramanujan’s Notebook, Part III, Springer-Verlag, New York, (1991). pp. 01- 72.

2. G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook and other unpublished papers, Norosa Publishing House, New Delhi, (1988), pp.1-120.

3. G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook, Amer. Math. Monthly, 86, (1979), pp.89-108

4. G.H. Hardy, and E.M. Wright, Introduction to the Theory of Numbers, 4th Edition, Oxford, Clarendon Press, (1965).

5. MacMhon, Combinatory analysis, Ann Arbor, Michigan: University of Michigan Library (2005).