Pythagorean Triples

Pythagorean triples — integer solutions to a² + b² = c² — have been studied for at least 4,000 years. The Plimpton 322 clay tablet (Babylonian, c. 1800 BCE) lists 15 Pythagorean triples, some involving very large numbers such as (13500, 12709, 18541). Euclid's formula generates all primitive triples: a = m² − n², b = 2mn, c = m² + n² for coprime m > n. The 3-4-5 triple was used by ancient Egyptian builders to create right angles: knot a rope into 12 equal segments, form a triangle with sides 3, 4, and 5, and the corner between 3 and 4 is exactly 90°.