A Non-Existence Property of Pythagorean Triangles with a 3-D Application

TL;DR

This paper proves the non-existence of certain Pythagorean triangle configurations and Pythagorean boxes using number theory and diophantine equations, with applications to 3D geometric structures.

Contribution

It introduces new non-existence results for specific Pythagorean triangles and boxes, employing novel propositions and diophantine equation analysis.

Findings

No pairs of Pythagorean triangles with specified hypotenuse-leg relationships exist.

Certain Pythagorean boxes with specific face and diagonal properties do not exist.

Proves key diophantine equations have no positive integer solutions.

Abstract

After the introduction, in section 2 we state the well known parametric formulas that describe the entire family of Pythagorean triples. In section 3, we list four well known results from number theory, used later in the paper. in section 2, we prove two propositions. Proposition 1 says that the diophantine equation z^2=x^4+4y^4, has no solutions in positive integers x,y, and z. We then use Proposition 1, to prove the insolvability of a certain four-variable diophantine system, which is Proposition 2. In Section 5, we present some examplesof pairs of Pythagorean triangles with a common hypotenuse. In Section 6, we use Proposition 2 to prove Theorem 1: There exists no pair of Pythagorean triangles such that the leg of largest length in the first triangle is the hypotenuse of the second triangle; and in addition, with the leg of smallest length in the first triangle; having the same length as a leg in the second triangle. in Section 7, we present information on Pythagorean boxes, and we describe how to generate infinitely many Pythagorean boxes with a face diagonal of integral length. Finally, in Section 8, we apply Proposition 2 to prove Theorem 2: There exists no Pythagorean box with a pair of congruent (or opposite) faces being squares. And with the four diagonals of equal length in another pair of opposite faces, also having integral length.