On generalized Mersenne Primes and class-numbers of equivalent quadratic fields and cyclotomic fields

Abstract

In this paper we define equivalent quadratic fields and prove that generalized Mersenne primes generate a family of infinitely many equivalent quadratic fields with equivalent index \(2\) and whose class numbers are divisible by 3. We also prove that the class-number of the cyclotomic field \(\mathbb {Q}\big ( \zeta _m \big )\), where \(m\in \mathbb {N}\) and \(\zeta _m\) is a primitive \(m\)-th root of unity, is divisible by a certain integer \(g\).