Show that it doesn't exist any of natural number $ n = 4m + 3$ that $ n= x^2+y^2 $ for any natural x and y Show that it doesn't exist any of natural number  $ n = 4m + 3$ that $  n= x^2+y^2 $ for any natural x and y
Show that every prime number in form $ p=4m+1 $ could be showed as $ p = x^2+y^2$ (x and y are natural)
I checked it and it

For the first question, any square number modulo four is 0 or 1 [even ==> 0, odd ==> 1].  This can be shown rather easily.  Then, the summation of two such numbers would never equal 3 modulo four showing the result.
