Primes And Square Sums
Prove that there exists no prime which is one less than a multiple of four that can be written as the sum of two squares.
Let p = a2 + b2.
With the exception of p = 2 (which is not one less than a multiple of 4 anyway) all primes are odd. Hence one square must be even and one square must be odd for their sum to be odd.
So W.L.O.G. (without loss of generality) let a = 2x (even) and
b = 2y + 1 (odd).
a2 = (2x)2 = 4x2
b2 = (2y + 1)2 = 4y2 + 4y + 1
p = 4x2 + 4y2 + 4y + 1 = 4(x2 + y2 + y) + 1 = 4k + 1
That is, an odd square and an even square always adds to a number which is one more than a multiple of 4. Hence we prove that no prime which is one less than a multiple of 4 can be written as the sum of two squares.
It is important to realise that the proof given shows that an odd square added to an even square will produce a number of the form 4k + 1. It does not show that every number of the form 4k + 1 can be obtained by adding an odd and even square. For example, the number 9 is of the form 4k + 1, and this cannot be obtained by adding two squares. However, Pierre de Fermat (1601-1665) was able to prove that ALL primes of the form 4k + 1 can be written as the sum of two squares, but the proof for this is far beyond the scope of an elementary approach.