Ramanujan’s Congruences and Identities

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


By inspecting a list of values of þ (n) for n up to 200, Ramanujan came to the conjecture that
$$\eqalign{ & p(5n + 4) \equiv 0(\bmod 5), \cr & p(7n + 5) \equiv 0(\bmod 7), \cr} $$
$$p(11n + 6) \equiv 0(\bmod 11).$$
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
$$p(243) \equiv 245 \equiv 0(\bmod {7^3})$$
$$p(243) \equiv 0(\bmod {7^3}).$$
Here λ = 243 and 24λ = 24.243 ≡ 1 (mod 73) .


Formal Power Series Recursion Formula Binomial Coefficient Arithmetical Statement Divisibility Property 
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© Springer-Verlag, Berlin • Heidelberg 1973

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  • Hans Rademacher

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