ABSTRACT
We shall now study the decomposition of positive integers into sums of integer squares. Theorems 7.8 and 7.12 will be needed later; the proofs may be omitted at a first reading.
We first ask: Which integers n have the form
n = x2 + y2
where x and y are integers? If
q = x2 + y2, r = z2 + t2,
then qr = (x2 + y2)(z2 + t2) = (xz + yt)2 + (xt− yz)2 . (7.1)
So:
LEMMA 7.1
If each prime factor of the positive integer n can be written as the sum of two integer squares, then n can be written in that way.
