Can Any Integer Congruent to 3 Modulo 4 Be Expressed as a Sum of Two Squares?

[pandaBee]

Homework Statement

All work is done in the set of integers, Z
a is congruent to 3 modulo 4, Want to prove a=/= c^2 + d^2 for any c or d

Homework Equations

The Attempt at a Solution

My work:
a congruent to 3 modulo 4 => a -3 = 4k
or a = 4k +3
Then want to prove that: 4k+3 =/= c^2 + d^2 *** for any c or dI've tested out some experimental values for 4k+3 and indeed none of them have any c or d that match this equation but I'm not sure how to progress from *** onwards, It is probably a very simple matter but I can't wrap my head around this.

 

[Dick]

Homework Statement

All work is done in the set of integers, Z
a is congruent to 3 modulo 4, Want to prove a=/= c^2 + d^2 for any c or d

Homework Equations

The Attempt at a Solution

My work:
a congruent to 3 modulo 4 => a -3 = 4k
or a = 4k +3
Then want to prove that: 4k+3 =/= c^2 + d^2 *** for any c or d


I've tested out some experimental values for 4k+3 and indeed none of them have any c or d that match this equation but I'm not sure how to progress from *** onwards, It is probably a very simple matter but I can't wrap my head around this.

You can express ANY integer c as 4k, 4k+1, 4k+2 or 4k+3. What are the possible expressions for c^2? In other words, what are the possible values of c^2 mod 4?