Babylonian Reciprocals and Dickson's Method for Pythagorean Triples — E8 Intelligence Research

FINDING: Babylonian reciprocal-pair algorithm generates Pythagorean triples via sexagesimal reciprocals (x, 1/x) scaled by a common factor, producing triples (p, q, d) as seen on Plimpton 322. Dickson's method (r, s, t) generalizes this: any triple (a, b, c) satisfies a = r + s, b = r + t, c = r + s + t, with r² = 2st. MATH: - Babylonian method: Choose regular sexagesimal numbers p, q (reciprocal pair, pq = 1 in base-60). Then triple sides: a = p - 1/p, b = 2, c = p + 1/p (scaled by common factor to clear denominators). Equivalent to modern: a = m² - n², b = 2mn, c = m² + n², with m = √p, n = 1/√p. - Dickson's method: For integers r, s, t with r² = 2st, then (r+s, r+t, r+s+t) is a Pythagorean triple. - Plimpton 322 triples correspond to r = 2, s = 1/x², t = x² (scaled), linking directly to reciprocal pairs. CONNECTION: - The reciprocal-pair method inherently uses base-60 regular numbers (smooth numbers with prime factors 2, 3, 5). This yields ratios between sides that approxi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com