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Singleton difference

Problem 136

Published on Friday, 29th December 2006, 06:00 pm; Solved by 3407; Difficulty rating: 50%

The positive integers, x, y, and z, are consecutive terms of an arithmetic progression. Given that n is a positive integer, the equation, x2y2z2 = n, has exactly one solution when n = 20:

132 − 102 − 72 = 20

In fact there are twenty-five values of n below one hundred for which the equation has a unique solution.

How many values of n less than fifty million have exactly one solution?