Proof of infinitely many solutions

  • Context: Undergrad 
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SUMMARY

The discussion centers on proving that there are infinitely many positive integer solutions to the equation x² + y² = z² using two methods: the first involves the parameterization x = m² - n², y = 2mn, z = m² + n², and the second uses x = 3m, y = 4m, z = 5m. Participants emphasize that by varying m and n, one can generate an infinite number of distinct solutions. The proof's validity hinges on demonstrating that different pairs of (m, n) yield unique triples (x, y, z), thus confirming the existence of infinitely many solutions.

PREREQUISITES
  • Understanding of Pythagorean triples
  • Familiarity with algebraic manipulation and factorization
  • Knowledge of integer parameterization techniques
  • Basic proof-writing skills in mathematics
NEXT STEPS
  • Study the derivation of Pythagorean triples using integer parameters
  • Learn about the uniqueness of solutions in Diophantine equations
  • Explore advanced proof techniques in number theory
  • Investigate the implications of parameterization in mathematical proofs
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Mathematics students, educators, and anyone interested in number theory and proof techniques, particularly those focusing on Pythagorean triples and integer solutions to equations.

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

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