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Ramanujan’s Famous Partition Congruences
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Volume 4, 2016
Issue 3 (June)
Pages: 17-22   |   Vol. 4, No. 3, June 2016   |   Follow on         
Paper in PDF Downloads: 67   Since Sep. 13, 2016 Views: 1489   Since Sep. 13, 2016
Md. Fazlee Hossain, Department of Mathematics, University of Chittagong, Chittagong, Bangladesh.
Nil Ratan Bhattacharjee, Department of Mathematics, University of Chittagong, Chittagong, Bangladesh.
Sabuj Das, Department of Mathematics, Raozan University College, Chittagong, Bangladesh.
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
G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook, Amer. Math. Monthly, 86, (1979), pp. 89-108.
G. E. Andrews, An Introduction to Ramanujan’s Lost Notebook and other unpublishedpapers, Norosa Publishing House, New Delhi, (1988), pp.1-120.
B. C. Berndt, Ramanujan’s Notebook, Part III, Springer-Verlag, New York, (1991).pp. 01-72.
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).
R. A. Rankin, Ramanujan’s manuscripts and notebooks II, Bull. London Math. Soc. 21 (1989), 351-365, reprinted in [2, pp. 129-142].
S. Ramanujan, Ramanujan’s hand writing letter to Hardy and sheets, (1916).
S. Ramanujan, Highly composite numbers, Proc. London Math. Soc. 14 (1915), 347-400.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, The Lost Notebook and Other unpublished Papers, Narosa, New Delhi, 1988.
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