What question did this study set out to answer?

The aim is to identify conditions under which a quadratic order is a unique factorization domain.

April 23, 2026International Journal of Mathematics and Mathematical Sciences0 citationsOpen Access

Criterion for Unique Factorization in Quadratic Orders

Key Points

The aim is to identify conditions under which a quadratic order is a unique factorization domain.
Analyzed quadratic orders within real quadratic fields.
Developed a criterion based on Diophantine equation solvability.
Investigated the role of positive fundamental discriminants and prime numbers.
Established a necessary and sufficient condition for unique factorization in quadratic orders.
Found a direct link between specific Diophantine equations and unique factorization domains.
Identified constraints related to the discriminant and its associated prime.

Abstract

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.

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