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6. Prove that the diophantine equation 3x² + 4y2 = 5z² has no non-trivial (i.e. (x, y, z) # (0,0,0)) solutions. [Give a proof by contradiction. Obtain a contradiction by proving that if there is a non-trivial solution then there is a solution (x₁, y₁, Z₁) with x₁ # 0 mod 5 or y₁ # 0 mod 5.] Deduce that the equation 3x² + 4y² = 5 has no rational solutions.

6. Prove that the diophantine equation 3x² + 4y2 = 5z² has no non-trivial (i.e. (x, y, z) # (0,0,0)) solutions. [Give a proof by contradiction. Obtain a contradiction by proving that if there is a non-trivial solution then there is a solution (x₁, y₁, Z₁) with x₁ # 0 mod 5 or y₁ # 0 mod 5.] Deduce that the equation 3x² + 4y² = 5 has no rational solutions.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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6. Prove that the diophantine equation 3x² + 4y² = 5z² has no non-trivial (i.e. (x, y, z) # (0,0,0)) solutions. [Give a proof by contradiction. Obtain a contradiction by proving that if there is a non-trivial solution then there is a solution (x₁, y₁, Z₁) with x₁ # 0 mod 5 or y₁ # 0 mod 5.] Deduce that the equation 3x² + 4y² = 5 has no rational solutions.
Transcribed Image Text:6. Prove that the diophantine equation 3x² + 4y² = 5z² has no non-trivial (i.e. (x, y, z) # (0,0,0)) solutions. [Give a proof by contradiction. Obtain a contradiction by proving that if there is a non-trivial solution then there is a solution (x₁, y₁, Z₁) with x₁ # 0 mod 5 or y₁ # 0 mod 5.] Deduce that the equation 3x² + 4y² = 5 has no rational solutions.
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