Question

Find the necessary and sufficient condition that a prime is expressible as sum of two squares

Answer 1

Step-by-step explanation:

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

p = x 2 + y 2 , {\displaystyle p=x^{2}+y^{2},} {\displaystyle p=x^{2}+y^{2},}

with x and y integers, if and only if

p ≡ 1 ( mod 4 ) . {\displaystyle p\equiv 1{\pmod {4}}.} p \equiv 1 \pmod{4}.

The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways:

5 = 1 2 + 2 2 , 13 = 2 2 + 3 2 , 17 = 1 2 + 4 2 , 29 = 2 2 + 5 2 , 37 = 1 2 + 6 2 , 41 = 4 2 + 5 2 . {\displaystyle 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad 17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},\quad 37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}.} 5 = 1^2 + 2^2, \quad 13 = 2^2 + 3^2, \quad 17 = 1^2 + 4^2, \quad 29 = 2^2 + 5^2, \quad 37 = 1^2 + 6^2, \quad 41 = 4^2 + 5^2.