Criterion for Unique Factorization in Quadratic Orders

This paper addresses the problem of unique factorization in quadratic orders within real quadratic fields. We establish a necessary and sufficient condition for a quadratic order , where Δ is a positive fundamental discriminant, to be a unique factorization domain (UFD). Our criterion is based on the solvability of certain Diophantine equations of the form |x 2 − dy 2 | = c 2 p, where p is a prime number satisfying specific conditions related to Δ.