Dorin Andrica, Vlad Crisan: Applications of the class number one problem to Diophantine equations and divisibility problems , p. 19-24

Abstract:

We investigate the properties of positive definite binary quadratic forms in order to solve a particular class of Diophantine equations of the form $ax^2+bxy+cy^2=n$. We also make some observations on the primes $p$ which divide expressions of the form $u\cdot n^2+v\cdot n+t$, with $u,v,t \in \Z$, subject to some constraints.

Key Words: Binary quadratic forms, Diophantine equations, divisibility

2010 Mathematics Subject Classification: Primary 11D09,
Secondary 11D45, 11A51