A theorem of Ramanujan concerning binary quadratic forms
Let \(a\), \(b\), \(A\), \(B\) be positive integers such that (i) \((a,b)=(A,B)=1\), (ii) \(ab\) is not a square, and (iii) every prime \(p\equiv B\pmod A\) with \(p\nmid 2ab\) is expressible in the form \(p=ax^2-by^2\) for some integers \(x\) and \(y\). Ramanujan asserted in his notebook that every prime \(q\equiv -B\pmod A\) with \(q\nmid 2ab\) is expressible in the form \(q=bu^2- av^2\) for some integers \(u\) and \(v\). This was first proved by the reviewer [J. Number Theory 38, 118--133 (1991; Zbl 0724.11016)] using Gauss' theory of genera of binary quadratic forms. In this paper the author gives a proof using class field theory of a more general result.
scientific article; zbMATH DE number 431320