# Sum of two squares

1. Jan 1, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
Prove that no prime three more than a multiple of four is a sum of two squares. (Hint: Work modulo 4.)

2. Relevant equations

3. The attempt at a solution

a^2+b^2=4n+3=3 mod 4 is impossible if you look at the possibilities of a^2 and b^2.

I did not use the fact that the number is prime. Am I missing something?

2. Jan 1, 2008

### Dick

Doesn't seem likely, does it?

3. Jan 1, 2008

### ehrenfest

What doesn't seem likely?

4. Jan 2, 2008

### Dick

"Am I missing something?" doesn't seem likely. You checked squares are 0 or 1 mod 4. So the sum of two of them doesn't equal 3. Looks pretty bulletproof to me.

5. Jan 2, 2008

### Defennder

What does "prime three" here mean?

6. Jan 2, 2008

### HallsofIvy

Staff Emeritus
It doesn't mean anything- but I did a double take myself when I read that. Perhaps commas would help: No prime, three more than a multiple of four, is a sum of two squares

7. Jan 2, 2008

### ehrenfest

In my defense, Loren Larson (who is all-knowing) didn't think this sentence required any commas.

Last edited: Jan 2, 2008