Pythagorean Triples in an AP
A Primitive Pythagorean triple is one whose three integers have a greatest common divisor of 1. How many Primitive Pythagorean triples are in an arithmetic progression ?
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1 solution
Let our three integers by x, y, and z. For this to be an arithmetic series, z - y = y - x. Rearranging, z = 2y - x. Plugging this into the Pythagorean theorem, x 2 + y 2 = 4 y 2 − 4 x y + x 2 . Rearranging, 3 y 2 = 4 x y and 3 y = 4 x . To get our relatively prime triple, we must substitute x and y for one (any other coprime integers would make x and y too large). So, there is 1 such triple: 3 , 4 , 5 .