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

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.

💡 Research Summary

The paper investigates structural impossibility results concerning Pythagorean triangles and their three‑dimensional analogue, the so‑called Pythagorean box (a rectangular prism whose faces are right‑angled triangles with integer side lengths). After a brief introduction, Section 2 recalls the classical parametrisation of all primitive and non‑primitive Pythagorean triples: for integers (m>n>0) with (\gcd(m,n)=1) and opposite parity, the legs and hypotenuse are given by (a=m^{2}-n^{2},;b=2mn,;c=m^{2}+n^{2}). Section 3 lists four elementary number‑theoretic facts that will be used later, including basic modular properties of squares, the behavior of fourth powers modulo 16, and the Euclidean algorithm for greatest common divisors.

Section 4 contains the two central propositions. Proposition 1 asserts that the Diophantine equation
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