Some results on bipartitions with 3-core

In this paper, we investigate the arithmetic properties of bipartitions with 3-core. Let A3(n) denote the number of bipartitions with 3-core of n. We will prove one infinite family of congruences modulo 5 for A3(n). We also establish one surprising congruence modulo 14 for A3(8n+6). Finally, we prove that, if u(n) denotes the number of representations of a nonnegative integer n in the form x2+y2+3z2+3t2 with x,y,z,t∈Z, then u(6n+5)=12A3(2n+1).