Divisibility of class numbers of imaginary quadratic fields whose discriminant has only three prime factors (Q2448933)

It is nowadays rather easy to construct quadratic number fields whose class number is divisible by a given integer \(n > 1\). In this article, a much deeper problem is solved: the author shows that there exist infinitely many complex quadratic number fields whose discriminant has exactly three prime factors and whose class group contains an ideal class of order divisible by a given integer \(n>1\). The algebraic part of the proof is the simple lemma that if \(d\) is squarefree of the form \(d = 4m^{2\ell} - n^2\), where \(n\) is odd and \(2m^\ell-n > 1\), then the class group of \({\mathbb Q}(\sqrt{-d}\,)\) contains a class of order \(2\ell\). The main part of the proof is analytic and uses the circle method for estimating the sum \(\sum \ell m^{\ell-1}\) over all primes \(p_1\), \(p_2\), \(p_3\) with \(p_1 + p_2p_3 = 4m^\ell\).