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DIOPHANTINE EQUATIONS A #Pythagorean triple is a set of three integers x, y, z such that x 2 + y2 = z2; the triple is said to be primitive if gcd(x, y, z) = 1. x 2 + y2 = z2 Because the length z of the hypotenuse of a right triangle is related to the lengths x and y of the sides by the famous Pythagorean equation x 2 + y2 = z2, the search for all positive integers that satisfy Eq. (1) is equivalent to the problem of finding all right triangles with sides of integral length. The latter problem was raised in the days of the Babylonians and was a favorite with the ancient Greek geometers. Pythagoras himself has been credited with a formula for infinitely many such triangles, namely, x = 2n + 1 y = 2n2 + 2n z = 2n2 + 2n + 1 Verify that 3, 4, 5 is the only primitive Pythagorean triple involving consecutive positive integers.