Sum of Three Squares
No integer of the form 4**n(8m+7) can be represented as the sum of three squares
Proof:
a==0 or 1 [mod 2]
a**2==0, 1 or 4 [mod 8]
a**2+b**2+c**2==0,1,2,3,4,5,or 6 [mod 8]
=> 8m+7 not equal to sum of three squares
a==0 or 1 [mod 2]
a**2==0 or 1 [mod 4]
a**2+b**2+c**2==0 [mod 4] => a==b==c==0 [mod 2]
So, 4**n(8m+7) = a**2+b**2+c**2 => a=2a1, b=2b1, c=2c1
and 4**(n-1)(8m+7) = a1**2+b1**2+c1**2
Mathematical induction completes the proof