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Solving a three-variable Diophantine

  1. Jun 25, 2012 #1
    I have the following equation

    $$(4x^2+1)(4y^2+1) = (4z^2+1)$$

    For positive, nonzero integers x and y (and thus z). I am having difficulty figuring out a good method/algorithm for calculating solutions to this equation. Any thoughts?
     
  2. jcsd
  3. Jun 27, 2012 #2
    One nice solution: x=56, y=209, z=23409
     
  4. Jun 27, 2012 #3
    A 'good deal' of the solutions are caught by:

    let x be a natural number and y = [itex]4 x^{2}[/itex],

    then we have z = [itex]x (2 y +1)^{2}[/itex]

    for example (x,y,z) = (1,4,9), (2,16,66), (3,36,219), ..., (17,1156,39321)
     
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