No prime p of the form 4k+3 , (p≡3 (mod 4)) is a sum of two squares.

Published On :2021-06-24 13:33:00

 

[begin{array}{l}\ Modulo;4,;;we;have;four;possible;remainders;\\ a = 0,1,2,;;or;3;for;any;integer;a;;;consequently,;;{a^2} = 0;or;1;(mod4).\\ It;follows;that,;;for;arbitrary;integers;a;and;b,{a^2} + {b^2} = 0,;;1,or;2;(mod4).\\ ;Because;p = 3;(mod4),;the;equation;p = {a^2} + {b^2};is;impossible.\ end{array}]