The residue of \(p(N)\) modulo small primes (Q1265274)

Let \(l\) be a prime. A partition of \(n\) is called \(l\)-affine, \(l\)-square affine accordingly as each part is a power of \(l\), an even power of \(l\), respectively. Let \(C(t,n)\) denote the number of \(t\)-core partitions of \(n\). The author first proves that if \(n<l^{2s+2}\), then \(p(n)\) is congruent \(\pmod l\) to a sum of products \(\prod^s_{i=0} C(l,a_i)\), the sum taken over all \(l\)-square affine representations of \(n\), namely \[ n= \sum^s_{i=0} a_il^{2i}. \] He obtains more specific equivalent congruences in the cases \(l=3, 5, 7, 11\). One by-product is an alternate proof of the Ramanujan congruences \(\pmod{l^i}\), where \(l\in \{5, 7, 11\}\) and \(i\in \{1,2\}\).