Proof of infinitely many solutions

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The discussion focuses on proving that the equation x^2 + y^2 = z^2 has infinitely many positive integer solutions. The participants suggest using parameterizations, specifically x = m^2 - n^2, y = 2mn, and z = m^2 + n^2, to generate solutions by varying integers m and n. They emphasize the need to show that this method produces distinct solutions, noting that fixing n while varying m ensures unique triples (x, y, z). The conversation highlights the importance of rigor in proofs and the necessity of confirming that different values of m and n yield different solutions. Ultimately, the proof is established by demonstrating that the parameterization leads to an infinite number of distinct solutions.
  • #31
Im surprised I forgot about the uniqueness part.
 

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