Problem 66

Diophantine equation

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} – 13180^{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} – 22^{2} = 1

2^{2} – 31^{2} = 1

9^{2} – 54^{2} = 1

5^{2} – 62^{2} = 1

8^{2} – 73^{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.

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These problems are part of
Project Euler
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