Proving gcd(r,s)=1 with gcd(r^2-s^2, r^2+s^2)

  • Context: Undergrad 
  • Thread starter Thread starter awesome220
  • Start date Start date
  • Tags Tags
    Gcd
Click For Summary

Discussion Overview

The discussion revolves around proving that if gcd(r,s)=1, then gcd(r^2-s^2, r^2+s^2) equals either 1 or 2. The scope includes mathematical reasoning and exploration of properties of gcd.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the proof and seeks assistance.
  • Another participant proposes that if n divides both (r^2-s^2) and (r^2+s^2), then n must also divide 2r^2, suggesting a potential pathway to the solution.
  • A third participant questions how this reasoning leads to the conclusion that gcd(r^2-s^2, r^2+s^2) equals 1 or 2.
  • A later reply indicates that the participant may have understood the reasoning after further consideration.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants express confusion and seek clarification without resolving the proof definitively.

Contextual Notes

Participants do not fully articulate the assumptions or definitions that may affect the proof, and there are unresolved steps in the reasoning process.

awesome220
Messages
4
Reaction score
0
Can anyone help me with this?

If gcd(r,s)=1 then prove that gcd(r^2-s^2, r^2+s^2)=1 or 2.

i'm so confused.
 
Physics news on Phys.org
awesome220 said:
Can anyone help me with this?

If gcd(r,s)=1 then prove that gcd(r^2-s^2, r^2+s^2)=1 or 2.

i'm so confused.

Suppose n|(r^2-s^2) and n|(r^2+s^2). (This would be the case for the gcd of the two expressions.) Then there are some integers a, b with
an=r^2-s^2 and bn=r^2+s^2.
Then (a+b)n=2r^2 and so n divides 2r^2. Does this help?
 
I understand, but how does that give us that gcd (r^2-s^2, r^2+s^2) = 1 or 2?
 
nevermind, I think i see it! Thanks!
 

Similar threads

Undergrad Clarifications needed for an exercise in Hungerford's abstract algebra
  • · Replies 23 ·
Replies
23
Views
2K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Understanding the Results of gcd(x,n)
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad Equivalent definitons for primitive polynomials
  • · Replies 24 ·
Replies
24
Views
2K
MHB Properties of gcd's and relatively prime integers
  • · Replies 2 ·
Replies
2
Views
2K
MHB GCD of a and b: Proving $gcd(a',b')=1$
  • · Replies 4 ·
Replies
4
Views
2K
MHB Prove that CnXCm isomorphic to Cgcd(m,n)XClcm(m,n)
  • · Replies 3 ·
Replies
3
Views
3K
Can anyone please review/verify my proofs for gcd problem?
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Units in Z_m .... Anderson and Feil, Theorem 8.6 .... ....
  • · Replies 5 ·
Replies
5
Views
2K
MHB Units in Z_m .... Anderson and Feil, Theorem 8.6 .... ....
  • · Replies 2 ·
Replies
2
Views
2K