Show that a positive integer n is the sum of two squares iff 2n is the sum of two squares.
Solution
If n = a2 + b2, then 2n = (a+b)2 + (a-b)2. Conversely if 2n = a2 + b2, then n = ( (a+b)/2)2 + ( (a-b)/2)2, and since a2+b2 is even, so are a±b, and hence (a±b)/2 are integers.
© John Scholes
jscholes@kalva.demon.co.uk
1 Nov 2003
Last corrected/updated 1 Nov 03