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Some Restricted Partition Functions: Congruences Modulo 5

Published online by Cambridge University Press:  09 April 2009

D. B. Lahiri
D. B. Lahiri
Affiliation:
Indian Statistical InstituteCalcutta, India
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Ramanujan was the first mathematician to discover some of the arithmetical properties of p(n), the number of unrestricted partitions of n. His congruence,

for example, is famous [2; 3]. Some progress has been made since then; it is known that the congruence,

has an infinitude of solutions for any arbitrary value of r [4]. This is a somewhat weak relation, and one would have liked to obtain, if possible, stronger results of the type,

for ‘almost all’ values of n, which in its turn is derivable from another stronger relation, viz.,

also established by Ramanujan [2], where r(n) is Ramanujan's function defined by

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 9 , Issue 3-4 , May 1969 , pp. 424 - 432
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Dickson, L. E., History of the theory of numbers, vol. I (Chelsea Publishing Company, New York, 1952).Google Scholar
[2]Hardy, G. H., Ramanujan (Cambridge, 1940).Google Scholar
[3]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford, 4th ed. 1960).Google Scholar
[4]Newman, M., ‘Periodicity modulo m and divisibility properties of the partition function’. Trans. Amer. Math. Soc. 97 (1960), 225–236.Google Scholar
[5]Watson, G. N., ‘Über Ramanujansche Kongruenzeigenschaften der Zerfälungsanzahlen’, Math. Z. 39 (1935), 712–731.CrossRefGoogle Scholar