Ramanujan’s Famous Partition Congruences

DOI: https://doi.org/10.18034/ajase.v5i1.74

Authors

  • Md. Fazlee Hossain Professor, Department of Mathematics, University of Chittagong, Chittagong, BANGLADESH
  • Nil Ratan Bhattacharjee Professor, Department of Mathematics, University of Chittagong, Chittagong, BANGLADESH
  • Sabuj Das Senior Lecturer, Department of Mathematics, Raozan University College, Chittagong, BANGLADESH

Keywords:

Congruences
Enumerating
Modulo
Residues
Ramanujan’s Lost Notebook

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.

References

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

G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook and other unpublished papers, Norosa Publishing House, New Delhi, (1988), pp.1-120.

G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook, Amer. Math. Monthly, 86, (1979), pp.89-108 DOI: https://doi.org/10.1080/00029890.1979.11994743

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

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

S. Ramanujan, Ramanujan’s handwriting letter to Hardy and sheets, (1916).

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Published

2016-12-23

How to Cite

Hossain, M. F., Bhattacharjee, N. R., & Das, S. (2016). Ramanujan’s Famous Partition Congruences. Asian Journal of Applied Science and Engineering, 5(1), 145-158. https://doi.org/10.18034/ajase.v5i1.74

Issue

Vol. 5 No. 1 (2016)

Section

Articles

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Copyright (c) 2016 Asian Journal of Applied Science and Engineering

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