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Subtle Summed Squares (Posted on 2007-05-18) Difficulty: 3 of 5
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.

See The Solution Submitted by Old Original Oskar!    
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Solution Solution | Comment 1 of 6

It is sufficient to prove this for N=P*Q because then
N=P*R*S can be simplified by letting Q=P*S which has been proven.

let P = (a^2 + b^2) and Q = (c^2 + d^2)  with a>b>0  and d>c>0

P*Q = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2
= (ac + bd)^2 + (ad - bc)^2

  Posted by Jer on 2007-05-18 13:03:28
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