Use congruence mod 12 to show that all odd primes of the for

Use congruence mod 12 to show that all odd primes of the form x^2-3y^2 are of the form 12n+1.

Solution

x^2-3y^2

If x^2-3y^2 = odd prime say 3

Consider 12n+1

it will be like 13, 25, 37, 49.......

The prime numbers are 13, 37, ...

13 = 9+4 = 16-3(1) Thus proved for 13

37 = 1+36 = 49-3(2^2)

Next prime number is 61.

61 = 25+36 = 64-3(1^2)

Thus we can prove that congruence mod 12 to show that all odd primes x^2-3y^2 are of the form 12n+1 and vice versa