Problem 390

Triangles with non rational sides and integral area

Consider the triangle with sides 5, 65 and 68. It can be shown that this triangle has area 9.

S(n) is the sum of the areas of all triangles with sides (1+b^{2}), (1+c^{2}) and (b^{2}+c^{2}) (for positive integers b and c ) that have an integral area not exceeding n.

The example triangle has b=2 and c=8.

S(10^{6})=18018206.

Find S(10^{10}).

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