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

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. ...