Let p(n) denote the number of unrestricted partitions of n. It is known that p(5m+4), p(7m+5), and p(11m+6)≡0 (mod 5, 7, and 11 respectively).
Easy proofs of cases 5 and 7 are found in [1]. They depend on writing Π (1−xn)4 and Π(1−xn)6, (n>0), as double series, using well-known formulae of Euler and Jacobi.
The most elementary proof of case 11 seems to be found in [2], using different ideas. Another proof, which applies uniformly to cases 5, 7, 11, but is not elementary, is given in [3]. It is natural to ask for one along the same lines as for 5 and 7, i.e., using Π(1−xn)10 expressed as a double series.
