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A CONJECTURE OF MERCA ON CONGRUENCES MODULO POWERS OF 2 FOR PARTITIONS INTO DISTINCT PARTS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 27 March 2023, pp. 26-36
- Print publication:
- February 2024
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- Article
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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. Şt. Univ. Ovidius Constanţa 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 
$Q(n)$.
