A CONJECTURE OF MERCA ON CONGRUENCES MODULO POWERS OF 2 FOR PARTITIONS INTO DISTINCT PARTS

Let Q(n) denote the number of partitions of n into distinct parts. Merca ['Ramanujan-type congruences modulo 4 for partitions into distinct parts', An. St. Univ. Ovidius Constan,ta 30(3) (2022), 185-199] derived some congruences modulo 4 and 8 for Q(n) and posed a conjecture on congruences modulo powers of 2 enjoyed by Q(n). We present an approach which can be used to prove a family of internal congruence relations modulo powers of 2 concerning Q(n). As an immediate consequence, we not only prove Merca's conjecture, but also derive many internal congruences modulo powers of 2 satisfied by Q(n). Moreover, we establish an infinite family of congruence relations modulo 4 for Q(n).