# Big sum of squares

$\large 4\:475\:137 = 2016^2+641^2 \\ \large 4\:627\:633 = 1977^2+848^2$ The two large numbers above are prime numbers. Consider their product, $N = 4\:475\:137 \times 4\:627\:633 = 20\:709\:291\:660\:721$

Consider all possible ways in which $$N$$ can be written as the sum of two squares: $N = a^2 + b^2,\ \ \ a \geq b \geq 1.$

What is the average value of the possible values of $$a$$? If you think $$N$$ cannot be written as the sum of two squares, type 999.

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