Suppose a number N can be written as P times Q times R times..., where all of P, Q, R... can each be written as the sum of two perfect nonzero squares.
Show that in this case N itself can also be written as the sum of two perfect squares.
Start with (a²+b²) times (c²+d²)= (ac-bd)² + (ad+bc)².
We can recursively apply this formula to P and Q, and replace them both by a single new factor, also the sum of two perfect squares; then apply the formula again to this number and R, to replace them both by a new sum of two squares, and so on.
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