Prime and Perfect Square

For all primes p ≠2, (a,b ∈Z)

p= a² + b² <=> p ≡ 1 mod 4


(2=1² + 1²)
5= 1² + 2² = 1 + 4 ≡ 1 mod 4
13= 2² + 3² = 4 + 9 ≡ 1 mod 4
17=1² + 4² = 1 + 16 ≡ 1 mod 4
29= 2² + 5² = 4 + 25 ≡ 1 mod 4
37= 1² + 6² = 1 +36 ≡ 1 mod 4

Notes:

1) Perfect squares (4, 9, 16, 25... ) ≡ 0 or ≡ 1 mod 4
2) Prime (4n+1) = a² + b²   (Euler took 7 yrs to prove)

3) Gauss expanded the proof to quadratic reciprocity (2 prime numbers p & q are linked by mod 4)