# General congruences modulo 5 and 7 for colour partitions

@article{Saikia2021GeneralCM,
title={General congruences modulo 5 and 7 for colour partitions},
author={Nipen Saikia and Chayanika Boruah},
journal={The Journal of Analysis},
year={2021},
pages={1-10}
}
• Published 14 August 2020
• Mathematics
• The Journal of Analysis
For any positive integers n and r , let $$p_r(n)$$ p r ( n ) denotes the number of partitions of n where each part has r distinct colours. Many authors studied the partition function $$p_r(n)$$ p r ( n ) for particular values of r . In this paper, we prove some general congruences modulo 5 and 7 for the colour partition function $$p_r(n)$$ p r ( n ) by considering some general values of r . To prove the congruences we employ some q -series identities which are also in the spirit of Ramanujan.
2 Citations

## References

SHOWING 1-10 OF 21 REFERENCES
Infinite families of congruences modulo 7 for Ramanujan’s general partition function
• Mathematics
• 2018
For any non-negative integer n and non-zero integer r, let $$p_r(n)$$pr(n) denote Ramanujan’s general partition function. In this paper, we prove many infinite families of congruences modulo 7 for
Distribution of the partition function modulo $m$
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin,
Infinite Families of Congruences for 3-Regular Partitions with Distinct Odd Parts
• N. Saikia
• Mathematics
Communications in Mathematics and Statistics
• 2019
Let $$pod_3(n)$$pod3(n) denote the number of 3-regular partitions with distinct odd parts (and even parts are unrestricted) of a non-negative integer n. In this paper, we present infinite families of
Ramanujan Congruences for p-k (n)
• A. Atkin
• Mathematics
• 1968
Let 1 2 Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these
Partitions in 3 colours
We study $$p_3(n)$$p3(n), the number of partitions of n in three colours, and derive congruences for $$p_3(n)$$p3(n) modulo high powers of 3.
A SIMPLE PROOF OF WATSON'S PARTITION CONGRUENCES FOR POWERS OF 7
Abstract Ramanujan conjectured that if n is of a specific form then p(n), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture
SOME CONGRUENCES DEDUCIBLE FROM RAMANUJAN'S CUBIC CONTINUED FRACTION
• Mathematics
• 2011
We present some interesting Ramanujan-type congruences for some partition functions arising from Ramanujan's cubic continued fraction. One of our results states that if p3(n) is defined by \$\sum_{n =
Identities and Congruences of the Ramanujan Type
Let P(n) denote the number of unrestricted partitions of the positive integer n. Ramanujan conjectured that (1.1) if . He also indicated that such congruences could be deduced from identities of the
Collected Papers
THIS volume is the first to be produced of the projected nine volumes of the collected papers of the late Prof. H. A. Lorentz. It contains a number of papersnineteen in all, mainly printed
Ramanujan: Letters and Commentary
• Economics
• 1995
A brief biography of Ramanujan Ramanujan in Madras (chapter 1) Ramanujan's first two letters to Hardy and Hardy's response (chapter 2) Preparing to go to England (chapter 3) Ramanujan at Cambridge