Proof of infinitely many solutions

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    Proof
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Discussion Overview

The discussion revolves around proving that there are infinitely many solutions in positive integers x, y, and z to the equation x² + y² = z². Participants explore various methods of proof, including parameterizations and algebraic manipulations, while seeking clarity on how to demonstrate the infinitude of solutions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant suggests using the parameterization x = m² - n², y = 2mn, z = m² + n² to generate solutions.
  • Another participant emphasizes the need to show that this method produces infinitely many distinct x values.
  • Some participants discuss the implications of having x, y, and z with no common factors, complicating the proof.
  • There are suggestions to reduce the final equation to an absolute truth to demonstrate the existence of infinite solutions.
  • Several participants express uncertainty about the correct steps to finalize the proof and the meaning of "end terms" in the context of the equation.
  • One participant notes that the proof should ultimately show equality without reducing it to trivial statements like 2 = 2.
  • There is a focus on ensuring that the construction of z aligns with the derived equations to complete the proof correctly.

Areas of Agreement / Disagreement

Participants generally agree on the approach of using parameterizations to find solutions, but there is no consensus on the best method to conclusively prove the infinitude of solutions. The discussion remains unresolved regarding the final steps of the proof and the clarity of the argumentation.

Contextual Notes

Participants express confusion over specific algebraic manipulations and the significance of certain terms in the proof. There is also mention of the necessity for m to be greater than n to ensure positive values for x.

  • #31
Im surprised I forgot about the uniqueness part.
 

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