Partition function (number theory): Difference between revisions

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==Congruences==

Mathematician Srinivasa Ramanujan is credited with discovering that "congruences" in the number of partitions exist for integers ending in 4 and 9. Starting at the number 4, the partition number for every 5th integer is divisible by 5. For instance, the number of partitions for the integer 4 is 5. For the integer 9, the number of partitions is 30; for 14 there are 135 partitions. The integers 5, 30, and 135 are all evenly divisabledivisible by 5. RamanjuanRamanujan demonstrated that this congruence goes indefinitely. He also discovered congruences related to 7 and 11. Starting with 5, the partition number for every 7th integer (5, 12, 19, ... ) is itself divisible by 7. Starting with 6, the partition number for every 11th integer is divisible by 11. Since 5, 7, and 11 are consecutive primes, one might think that starting with 7, the partition number of every 13th integer is divisible by 13. This is however, quite false.

In more recent times, A. O. L. Atkin of the University of Illinois at Chicago discovered additional congruences. For example, the partition number for every 17,303rd integer starting with 237 is divisible by 13.