Rational $\theta$-parallelogram envelopes via $\theta$-congruent elliptic curves

TL;DR

This paper generalizes the concept of $ heta$-congruent numbers to rational $ heta$-parallelogram envelopes, using algebraic curves, extending previous work limited to right angles and addressing open questions for Pythagorean angles.

Contribution

It introduces a new generalization of $ heta$-congruent numbers, studies their properties via algebraic curves, and answers open questions for Pythagorean angles.

Findings

Generalization of $ heta$-congruent numbers to rational $ heta$-parallelogram envelopes.

Use of algebraic curve arithmetic to analyze these envelopes.

Extension of previous results to all Pythagorean angles.

Abstract

We introduce a new generalization of $\theta$-congruent numbers by defining the notion of rational $\theta$-parallelogram envelope for a positive integer $n$, where $\theta \in (0, \pi)$ is an angle with rational cosine. Then, we study more closely some problems related to the rational $\theta$-parallelogram envelopes, using the arithmetic of algebraic curves. Our results generalize the recent work of T.~Ochiai, where only the case $\theta=\pi/2$ was considered. Moreover, we answer the open questions in his paper and their generalizations for any Pythagorean angle.