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

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Homework Help Overview

The discussion revolves around proving that no prime number that is three more than a multiple of four can be expressed as a sum of two squares. The problem hints at using modular arithmetic, specifically modulo 4, to explore this assertion.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the implications of working modulo 4 and the nature of squares in that context. There is a question regarding the interpretation of the phrase "prime three more than a multiple of four," with some participants suggesting that clarity in punctuation could aid understanding.

Discussion Status

The discussion is active, with participants exploring the reasoning behind the assertion and questioning the clarity of the problem statement. Some participants express confidence in the reasoning presented, while others seek clarification on specific terms used in the problem.

Contextual Notes

There is a hint provided in the original problem to work modulo 4, and participants are considering the implications of this hint in their reasoning. The discussion also reflects on the phrasing of the problem, which may lead to confusion.

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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?
 
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Doesn't seem likely, does it?
 
Dick said:
Doesn't seem likely, does it?

What doesn't seem likely?
 
"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.
 
ehrenfest said:

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?
 
Defennnder said:
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
 
HallsofIvy said:
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.
 
Last edited:

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