Ramanujan's congruences

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In mathematics, Ramanujan's congruences refer to some remarkable congruences for the partition function p(n). The Indian mathematician Srinivasa Ramanujan discovered the following

  • p(5k+4)\equiv 0 \pmod 5
  • p(7k+5)\equiv 0 \pmod 7
  • p(11k+6)\equiv 0 \pmod {11}

In his 1919 paper (Ramanujan, 1919), he gave proof for the first two congruences using the following identities (using q-series notation):

\sum_{k=0}^\infty p(5k+4)q^k=5\frac{(q^5)_\infty^5}{(q)_\infty^6}
\sum_{k=0}^\infty p(7k+5)q^k=7\frac{(q^7)_\infty^3}{(q)_\infty^4}+49q\frac{(q^7)_\infty^7}{(q)_\infty^8}

After Ramanujan died in 1920, G. H. Hardy, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921). The proof in this manuscript employs Eisenstein series.

[edit] See also

[edit] References

  • S. Ramanujan, Some properties of p(n); the number of partitions of n, Proc. Cambridge Phi-los. Soc. 19 (1919), 207-210.
  • S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147-153.
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