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

Key Points

• 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.