Question

Shhow that prime number of the form 4n+3 can not be expressed as the sum of two squares.
Please dont give any silly answer . Because of silly answers I will not get right answer.

Answer 1

Proof: a2−1=(a−1)(a+1). Since a is odd, both a−1 and a+1 are even, so that a2−1 is divisible by 4. ■

Lemma 2: a
is even ⟹ a2≡0(mod4)

.

Proof: Trivial. ■

Now, suppose that u=a2+b2

.

(1) If both a
 and b are even, then u

is divisible by four by lemma 2.

(2) If both a
and b are odd, then u≡2(mod4)

by lemma 1.

(3) If a
 is even and b is odd (wlog), then u≡1(mod4)

by lemmas 1 and 2.

That is, it is never the case that u≡3(mod4)
.

Answer 2

Heya!!!

Here's ur answer..

Hope it may help you...

Thank you :) :))