# Regular ternary polygonal forms

## TL;DR

This paper proves that primitive regular m-gonal forms do not exist for large m by constructing prime sequences with specific properties related to quadratic fields.

## Contribution

It introduces a novel method of constructing prime sequences inert in quadratic fields to establish non-existence results for regular m-gonal forms.

## Key findings

- No primitive regular m-gonal forms for sufficiently large m
- Constructed prime sequences satisfy key inequalities
- Linked prime inertness to form regularity

## Abstract

Inspired by Dickson's classification of regular ternary quadratic forms, we prove that there are no primitive regular $m$-gonal forms when $m$ is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next prime by the product of the previous primes, a question of independent interest.