Some Results Connected with the Class Number Problem in Real Quadratic Fields

Abstract

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h(d) of the real quadratic field \( \mathbb{Q}{\left( {{\sqrt d }} \right)} \). In particular, we give a new proof of the result of Hasse, asserting that in this case h(d) = 1 is possible only if d is of the form p, 2q or qr, where p, q, r are primes and q ≡ r ≡ 3(mod 4).