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

June 24, 2021

\begin{array}{l} M o d u l o 4, w e h a v e f o u r p o s s i b l e r e m a i n d e r s \\ a = 0, 1, 2, o r 3 f o r a n y i n t e g e r a; c o n s e q u e n t l y, a^{2} = 0 o r 1 (m o d 4) . \\ I t f o l l o w s t h a t, f o r a r b i t r a r y i n t e g e r s a a n d b, a^{2} + b^{2} = 0, 1, o r 2 (m o d 4) . \\ B e c a u s e p = 3 (m o d 4), t h e e q u a t i o n p = a^{2} + b^{2} i s i m p o s s i b l e . \end{array}