Inequalities for inert primes and their applications

TL;DR

This paper establishes inequalities for inert primes related to a non-square integer D and applies these results to simplify the classification of irregular ternary quadratic forms.

Contribution

It introduces Euclid's type inequalities for primes with a specific Kronecker symbol and provides a new criterion for irregularity in ternary quadratic forms.

Findings

Proved inequalities for primes with (D/q_i) = -1.

Derived a new criterion for irregular ternary quadratic forms.

Simplified existing classification methods.

Abstract

For any given non-square integer $ D\equiv 0,1 \pmod{4} $, we prove Euclid's type inequalities for the sequence $ \{q_{i}\} $ of all primes satisfying the Kronecker symbol $ (D/q_{i})=-1 $, $ i=1,2,\cdots, $ and give a new criterion on a ternary quadratic form to be irregular as an application, which simplifies Dickson and Jones's argument in the classification of regular ternary quadratic forms to some extent.