What question did this study set out to answer?

May 20, 2026Open Access

The Emergence of √2 and √3 from Primitive Pythagorean Triples via a Cotangent Half-Angle Function

Puntos clave

The aim is to demonstrate the emergence of √2 and √3 from primitive Pythagorean triples using a cotangent half-angle function.
Defined the cotangent half-angle function F(a, b, c) on all primitive Pythagorean triples (a, b, c) with a > b.
Proved the double bound 1 + √2 < F(a, b, c) < √2 + √3 for primitive Pythagorean triples.
Established that F(a, b, c) lies within a bounded range, proving 1 + √2 < F(a, b, c) < √2 + √3 for all primitive Pythagorean triples.

Resumen

This paper establishes a remarkable embedding of both √2 and √3 within the structure of primitive Pythagorean triples, revealed through the cotangent half-angle function F(a, b, c) = cot(½ arctan(c/(a + b))) = √2(c² + ab) + (a + b) / c , defined on every primitive Pythagorean triple (a, b, c) with legs a > b and hypotenuse c. We prove the sharp double bound 1 + √2 < F(a, b, c) < √2 + √3 for all primitive Pythagorean triples. ...

Leer artículo completoexternamente

Me gusta

Guardar

Ver artículo completo