Inequalities for inert primes and regular ternary polygonal forms

Abstract

In 2015, Chan and Ricci proved that for each $ m\ge 3 $, there are finitely many primitive regular ternary $ m $-gonal forms, but we do not know a classification, or even explicit numbers of regular ternary $ m $-gonal forms for given $ m $, except the classification for $ m=4 $ that was completed by Dickson, Jones and Pall as early as 1931, the list for $ m=3 $ which was recently determined by M. Kim and B.-K. Oh.

In this thesis, motivated by Dickson's work in excluding irregular ternary quadratic forms, we study Euclid and Bonse-type inequalities for the sequences $ \{q_{i}\}_{D} $ of all primes satisfying the Kronecker symbol $ (D/q_{i})=-1 $, $ i=1,2,\cdots$, where $ D $ is the non-square discriminant of the quadratic field $ \mathbb{Q}(\sqrt{D}) $, using elementary and analytic methods. Also, as a simple application of these prime inequalities, we give a new criterion based on these inequalities that implies that a ternary quadratic form is irregular, which simplifies Dickson and Jones's argument in the classification of regular ternary quadratic forms to some extent. Inspired by that, we construct an analogous prime inequality involving $ \{q_{i}\}_{D} $ with additional restrictions by modifying Earnest's trick, which can be applied to the classification for regular ternary $ m $-gonal forms. Precisely, we deduce that there are no primitive regular ternary $ m $-gonal forms when $ m $ is sufficiently large, which improves Chan and Ricci's finiteness result of regular ternary $ m $-gonal forms.