Prove: No Prime 3+4n is Sum of 2 Squares

[ehrenfest]

Homework Statement

Prove that no prime three more than a multiple of four is a sum of two squares. (Hint: Work modulo 4.)

Homework Equations

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?

 

[Dick]

Doesn't seem likely, does it?

 

[ehrenfest]

Doesn't seem likely, does it?

What doesn't seem likely?

 

[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.

 

[Defennder]

Homework Statement

Prove that no prime three more than a multiple of four is a sum of two squares. (Hint: Work modulo 4.)

What does "prime three" here mean?

 

[HallsofIvy]

What does "prime three" here mean?

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

 

[ehrenfest]

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

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