## Diophantine equation

### Problem 66

Published on Friday, 26th March 2004, 06:00 pm; Solved by 11940; Difficulty rating: 25%Consider quadratic Diophantine equations of the form:

*x*^{2} – D*y*^{2} = 1

For example, when D=13, the minimal solution in *x* is 649^{2} – 13×180^{2} = 1.

It can be assumed that there are no solutions in positive integers when D is square.

By finding minimal solutions in *x* for D = {2, 3, 5, 6, 7}, we obtain the following:

3^{2} – 2×2^{2} = 1

2^{2} – 3×1^{2} = 1

9^{2} – 5×4^{2} = 1

5^{2} – 6×2^{2} = 1

8^{2} – 7×3^{2} = 1

Hence, by considering minimal solutions in *x* for D ≤ 7, the largest *x* is obtained when D=5.

Find the value of D ≤ 1000 in minimal solutions of *x* for which the largest value of *x* is obtained.