# Proof of a conjecture of Ramanujan

```@article{Atkin1967ProofOA,
title={Proof of a conjecture of Ramanujan},
author={A. O. L. Atkin},
journal={Glasgow Mathematical Journal},
year={1967},
volume={8},
pages={14 - 32}
}```
• A. Atkin
• Published 1 January 1967
• Mathematics
• Glasgow Mathematical Journal
We write and so that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n, and a suitably modified form for q = 7 and all n. (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically…
123 Citations
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## References

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Proof of Ramanujan’s partition congruence for the modulus 11³
Presented to the Society, October 30, 1948; received by the editors September 13, 1948 and, in revised form, February 9, 1949. 1 The author is greatly indebted to the referee, who made a very careful
Remarks on some modular identities
Introduction. We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup Fo(p) of the modular group r. By specializing these functions, we
Some properties of partitions. II
• Mathematics
• 1958
Let N(r,m, n) (resp. M(r,m, n)) denote the number of partitions of n whose ranks (resp. cranks) are congruent to r modulo m. Atkin and Swinnerton-Dyer gave the relations between the numbers
Congruence properties of the partition function and associated functions
• Mathematics
• 1952
The subject of this paper is the study of an unpublished manuscript by the late Srinivasa Ramanujan, the Indian mathematician. The manuscript covers forty-three pages of foolscap, and it is now in
Further Identities and Congruences for the Coefficients of Modular Forms
• M. Newman
• Mathematics