Consecutive Pythagorean triangle sides

In this post we find all Pythagorean triples that contain consecutive numbers, all Pythagorean triples (a, b, c) such that a + 1 = b or b + 1 = c. a + 1 = b George Osborne wrote a paper [1] addressing the question of when the squares of two consecutive numbers is also a square. Geometrically this is asking for primitive Pythagorean triples for which the legs are consecutive integers. He proved that the sequence shorter legs satisfies the recurrence relation with initial conditions u0 = 0 and u1 = 1. This is OEIS sequence A001652. The method for solving recurrences like the one above is analogous to the method for solving linear differential equations. See a solution here. This gives us the following formula for the terms: b + 1 = c It’s also possible for the longer side and hypotenuse of a Pythagorean triangle to be consecutive numbers, as in the (5, 12, 13) triangle. All primitive Pythagorean triples are given by Euclid’s formula with integers m > n > 0. If b and c are consecutive…