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Ramanujan’s Congruences and Identities

  • Hans Rademacher
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 169)

Abstract

By inspecting a list of values of þ (n) for n up to 200, Ramanujan came to the conjecture that
$$\eqalign{ & p(5n + 4) \equiv 0(\bmod 5), \cr & p(7n + 5) \equiv 0(\bmod 7), \cr} $$
(104.1)
$$p(11n + 6) \equiv 0(\bmod 11).$$
(104.2)
Ramanujan proved these congruences and also similar ones for the moduli 52, 72, 112, and came then to the general conjecture that if r = 57β11γ, and 24λ ≡ 1 (mod r) then
$$p(nr + \lambda ) \equiv 0(\bmod r).$$
In this form the statement is not true. Indeed S. Chowla observed, using a table of partitions prepared by Gupta that
$$p(243) \equiv 245 \equiv 0(\bmod {7^3})$$
but
$$p(243) \equiv 0(\bmod {7^3}).$$
Here λ = 243 and 24λ = 24.243 ≡ 1 (mod 73) .

Keywords

Formal Power Series Recursion Formula Binomial Coefficient Arithmetical Statement Divisibility Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, Berlin • Heidelberg 1973

Authors and Affiliations

  • Hans Rademacher

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