Prove that a positive integer n is the sum of two perfect squares if and only if 2*n is also the sum of two perfect squares.
Write the two squares as i^2 and (i+k)^2 where k >= 0
Then n = i^2 + i^2 + 2ik + k^2
= 2i^2 + 2ik + k^2
Twice that is:
4i^2 + 4ik + 2k^2
= (2i)^2 + 2(2i)k + k^2 + k^2
= (2i + k)^2 + k^2
which is also the sum of two squares
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Posted by Paul
on 2020-07-17 08:35:09 |
solution
