Pythagorean Triples are whole numbers that fit the famous formula of A[sup]2[/sup] + B[sup]2[/sup] = C[sup]2[/sup]. The lowest example being 3, 4, and 5. I was interested in triples where A and B differ by only 1. Other examples are 20,21, 29; 119, 120, 169; and 696, 697, 985. I found those examples through a formula* that does always yield triples where A and B differ by one; however, I don’t know if the formula gives all instances, or if there might be others that I’m missing. Any math wizs care to take a crack at this?
*the details would be three times longer than this post, but I’ll post it if anyone wants.
Mathworld has more information than you ever wanted on Pythagorean Triples. What you’re asking about are Twin Pythagorean Triples, specifically, what mathworld calls “leg-leg” twin Pythagorean triples.
I’ll have to read it in more detail later, but it looks like Pell numbers as applied to leg-leg Pythagorean triples is exactly what I was looking for. Thanks!
Ah! In general, I was using what I didn’t know were the Pell sequence numbers (0, 1, 2, 5, 12, 29, 70, etc.) to generate leg-leg Pythagorean triples. I knew that the terms of that series approximated √2 +1, but I didn’t have a justification for using only those numbers, or proof that all leg-leg triples would have to fit that sequence. Case closed.