Exceptional congruences for powers of the partition function

Congruences of the Partition Function

Ramanujan also conjectured that congruences (1) exist for the cases A = 5 , 7 , or 11 . This conjecture was proved by Watson [17] for the cases of powers of 5 and 7 and Atkin [3] for the cases of powers of 11. Since then, the problem of finding more examples of such congruences has attracted a great deal of attention. However, Ramanujan-type congruences appear to be very sparse. Prior to the la...

Extension of Ramanujan’s Congruences for the Partition Function modulo Powers of 5

Ramanujan Type Congruences for a Partition Function

We investigate the arithmetic properties of a certain function b(n) given by ∞ ∑ n=0 b(n)qn = (q; q)−2 ∞ (q 2; q2)−2 ∞ . One of our main results is b(9n + 7) ≡ 0 (mod 9).

A Simple Proof of Watson's Partition Congruences for Powers of 7

Ramanujan conjectured that if n is of a specific form then/>(«), the number of unrestricted partitions of n, is divisible by a high power of 7. A modified version of Ramanujan's conjecture was proved by G. N. Watson. In this paper we establish appropriate generating formulae, from which Watson's results follow easily. Our proofs are more straightforward than those of Watson. They are elementary...

Congruences for Andrews’ Smallest Parts Partition Function and New Congruences for Dyson’s Rank

Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain an explicit Ramanujan-type congruence for spt(n) ...