## Singleton difference

### Problem 136

Published on Friday, 29th December 2006, 06:00 pm; Solved by 3446; 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, *x*^{2} − *y*^{2} − *z*^{2} = *n*, has exactly one solution when *n* = 20:

13^{2} − 10^{2} − 7^{2} = 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?