The ratio of \theta-congruent numbers

TL;DR

This paper proves the existence of infinitely many pairs of heta-congruent numbers with a fixed ratio for any heta with rational cosine, generalizing previous results on classical congruent numbers.

Contribution

It extends the theory of congruent numbers by establishing the infinite existence of heta-congruent number pairs with prescribed ratios for arbitrary heta with rational cosine.

Findings

Infinitely many heta-congruent number pairs with ratio lN=kM exist.

Generalization from classical to arbitrary heta-congruent numbers.

Supports broader understanding of heta-congruent number properties.

Abstract

Let 0<\theta<\pi such that \cos\theta\in \Q. In this paper, we prove that for given positive square-free coprime integers k,l, there exist infinitely many pairs (M,N) of \theta-congruent numbers such that lN=kM. This generalize the previous result of Rajan and Ramaroson [A. Rajan and F. Ramaroson, Ratios of congruent numbers, Acta Arithemetica 128 (2007), no. 2, 101-106] on the ratio of congruent numbers from congruent numbers (i.e. \theta=\pi/2) to arbitrary \theta-congruent numbers.