Prove property of diophantine equation

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Discussion Overview

The discussion revolves around the solvability of the diophantine equation x² - y² = n in integers, specifically investigating the conditions under which this equation has integer solutions. The focus includes theoretical reasoning and mathematical justification.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant asserts that the equation is solvable if and only if n is odd or 4 divides n.
  • Another participant provides examples to illustrate the claim but is challenged on the adequacy of examples as proof.
  • A different participant suggests considering the equation modulo 4, arguing that it suffices to show that n = 2 mod 4 cannot occur.
  • Another participant proposes a substitution x = y + k, indicating that this approach may clarify the situation, though they express uncertainty about its effectiveness.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof of the claim. There are competing views on the adequacy of examples versus formal proof, and the discussion remains unresolved regarding the best approach to demonstrate the conditions for solvability.

Contextual Notes

Some assumptions about the properties of integers and modular arithmetic are implied but not explicitly stated. The discussion also reflects varying levels of rigor in the proposed arguments.

ascheras
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Show that the diophantine equation x^2 - y^2= n is solvable in integers iff n is odd or 4 divides n.
 
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ascheras said:
Show that the diophantine equation x^2 - y^2= n is solvable in integers iff n is odd or 4 divides n.

Well, 4^2-3^2=7 and 4^2-2^2=12, 12/4=3.
 
That isn't a proof. That is an example.

Consider the answer mod 4, one only needs to show n =2 mod 4 can't happen, which is straight forward.
 
And it becomes all the more obvious if you write x = y + k, for some integer k.

Edit : Well, maybe not...but it doesn't make it harder.
 
Last edited:

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