Vol. 28, No. 3, 1969

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Some restricted partition functions; Congruences modulo 3

D. B. Lahiri

Vol. 28 (1969), No. 3, 575–581
Abstract

We shall establish in this paper some congruence relations with respect to the modulus 3 for some restricted partition functions. The difference between the unrestricted partition function, p(n), and these restricted partition functions which we shall denote by

-rp(n) with r = 3,6,12,
27

merely lies in the restriction that no number of the forms 27n, or 27n ± r, shall be a part of the partitions which are of relevance in the restricted case. Thus to determine the value of r _ 27p(n) one should count all the unrestricted partitions of n excepting those which contain a number of any of the above forms as a part. We shall assume p(n) and r _ 27p(n) to be unity when n is zero, and vanishing when the argument is negative. We can now state our theorems.

Theorem 1. For almost all values of n

-3p(n) ≡ 6p(n) ≡ 27p(n) ≡ 0 (mod 3).
27       27      12

Theorem 2. For all values of n

3-       6-           27
27p(3n) ≡ 27p(3n + 1) ≡ −12p(3n+ 2)  (mod  3).

Mathematical Subject Classification
Primary: 10.48
Milestones
Received: 3 October 1967
Revised: 27 May 1968
Published: 1 March 1969
Authors
D. B. Lahiri