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

In summary, the conversation discusses how to prove that no prime number, three more than a multiple of four, can be expressed as the sum of two squares. The hint given is to work modulo 4 and consider the possibilities for the squares. The conversation concludes that the statement is true and does not require any commas for clarification.
  • #1
ehrenfest
2,020
1

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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  • #2
Doesn't seem likely, does it?
 
  • #3
Dick said:
Doesn't seem likely, does it?

What doesn't seem likely?
 
  • #4
"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
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?
 
  • #6
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
 
  • #7
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:

1. How do you prove that no prime number of the form 3+4n can be written as the sum of two squares?

To prove this statement, we will use a proof by contradiction. We will assume that there exists a prime number of the form 3+4n that can be written as the sum of two squares. Then, we will show that this assumption leads to a contradiction, thus proving that our original statement is true.

2. What is the significance of the form 3+4n in this statement?

The form 3+4n is significant because it represents all prime numbers that are congruent to 3 modulo 4. These numbers have a unique property that makes them unable to be written as the sum of two squares.

3. Can you provide an example of a prime number that cannot be written as the sum of two squares?

Yes, the prime number 7 cannot be written as the sum of two squares. This is because 7 is congruent to 3 modulo 4, and any number that is congruent to 3 modulo 4 cannot be written as the sum of two squares.

4. How does this proof relate to the Pythagorean theorem?

This proof uses the Pythagorean theorem in the form of a^2 + b^2 = c^2 to show that no prime number of the form 3+4n can be written as the sum of two squares. By assuming that a prime number of this form can be written as the sum of two squares, we are essentially saying that there exist integers a and b that satisfy the Pythagorean theorem. However, we will show that this is impossible, leading to a contradiction.

5. Is this statement true for all numbers, or just prime numbers?

This statement is true for all numbers of the form 3+4n, not just prime numbers. This is because the property that makes these numbers unable to be written as the sum of two squares is not related to their primality, but rather their congruence to 3 modulo 4.

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