New Recurrent Generators of Pythagorean Triples: Three Solutions Different from Euclid's Method

Abstract We present new recurrent generators for Pythagorean triples, i. e. , integer solutions of A² + B² = C². Three different families of recurrence formulas are given, each based on a fixed difference d = C - B (odd or even) or d' = B - A (odd). The first two families start from the well-known triple (3, 4, 5) and generate infinite series with constant C-B = 1 or 2. It is shown that these recurrences can be generalized to any d: if a triple with C-B = d is known, then all further triples with the same d are obtained by A₍+₁ = Aₙ + 2d, B₍+₁ = Aₙ + Bₙ + A₍+₁, C₍+₁ = Aₙ + Cₙ + A₍+₁. The method is illustrated with tables for d = 8, 9, 18, 25, 32, 49, 50. The third solution uses a different recurrence based on twice the sum A+B+C and covers the case B-A = 1 (and its multiples). The proposed recurrences provide an elementary, table‑friendly alternative to Euclid's parametric formula, generating all Pythagorean triples without missing any. The article contains 20+ numerical tables and a universal recurrent method.