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… 
Congruence Properties of Certain Restricted Partitions
It was proved by Ramanujan (see [1] and [2]) that the number p(n) of unrestricted partitions of a natural number n satisfies the following congruences: (1) p(n)_ 0 (mod 5) if n 4 (mod 5) (2) p(n)_ 0
SOME PROPERTIES OF p(n) AND c(n) MODULO POWERS OF 13
j(^r) = E c(n)xn = (1 +240 0 u3(n)xn) /xf24(x) 744, where 03(n) = d3. dln Then p(n) is just the number of unrestricted partitions of n, and c(n) is the Fourier coefficient of Klein's modular
Ramanujan Congruences for p-k (n)
  • A. Atkin
  • Mathematics
    Canadian Journal of 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
Congruences like Atkin's for the partition function
Let p(n) be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form p(Qln+ β) ≡ 0 (mod l) where l and Q are prime and 5 ≤ l ≤ 31; these lie in two
Distribution of the partition function modulo composite integers M
seem to be distinguished by the fact that they are exceptionally rare. Recently, Ono [O1] has gone some way towards quantifying the latter assertion. More recently, Ono [O2] has made great progress
A UNIFIED ALGORITHMIC FRAMEWORK FOR RAMANUJAN’S CONGRUENCES MODULO POWERS OF 5, 7, AND 11
In 1919 Ramanujan conjectured three infinite families of congruences for the partition function modulo powers of 5, 7, and 11. In 1938 Watson proved the 5-case and a corrected version of the 7-case.
Congruences Between Coefficients of a Class of Eta-Quotients and their Applications to Combinatorics
Ramanujan in 1920s discovered remarkable congruence properties of the partition function p(n). Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and
Congruences of multipartition functions modulo powers of primes
Let pr(n) denote the number of r-component multipartitions of n, and let Sγ,λ be the space spanned by η(24z)γϕ(24z), where η(z) is the Dedekind’s eta function and ϕ(z) is a holomorphic modular form
Construction of Modular Function Bases for Gamma0(121) related to p(11n + 6)
Motivated by arithmetic properties of partition numbers p(n), our goal is to find algorithmically a Ramanujan type identity of the form ∑∞ n=0 p(11n+6)qn = R, where R is a polynomial in products of
The spt-function of Andrews
TLDR
Recent developments in the study of spt($n), including congruence properties established by Andrews, Bringmann, Folsom, Garvan, Lovejoy and Ono et al., a constructive proof of the Andrews-Dyson-Rhoades conjecture, generalizations and variations of the spt-function, are summarized.
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References

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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
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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
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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
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  • M. Newman
  • Mathematics
    Canadian Journal of Mathematics
  • 1958
If n is a non-negative integer, define p r(n) by otherwise define p r(n) as 0. (Here and in what follows all sums will be extended from 0 to ∞ and all products from 1 to ∞ unless otherwise stated.) p
Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(τ)
The general idea behind the proof of this theorem is as follows. Consider the congruence (1. 2) as an example. By applying a certain linear operatordenoted by U5-to the right member of (1. 1) one
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Some Properties of Partitions
ON A SYSTEM OF MODULAR FUNCTIONS CONNECTED WITH THE RAMANUJAN IDENTITIES
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