Ramanujan's congruences

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In mathematics, Ramanujan's congruences are 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-Pochhammer symbol 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After Ramanujan died in 1920, G. H. Hardy, extracted proofs of all three congruences from an unpublished manuscript of Ramanujan on p(n) (Ramanujan, 1921).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.

In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruences modulo 11. Forty years later, George Andrews and Frank Garvan successfully found such a function, and proved the celebrated result that the crank simultaneously “explains” the three Ramanujan congruences modulo 5, 7 and 11.

Later Ken Ono conjectured that the elusive crank satisfies exactly the same types of general congruences as the partition function. This was proved by Karl Mahlburg in his 2005 paper Partition Congruences and the Andrews-Garvan-Dyson Crank, linked below. This paper won the first Proceedings of the National Academy of Sciences Paper of the Year prize.

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