Mathematics / Mathematik / Matemática

Mathematics / Mathematik / Matemática

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  • Winfried Weber
    Winfried Weber    Group moderator
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    Diophantic biquadratic function
    I would like to discuss the following diophantic biquadratic equations :

    Let a,b,c ≠0 three (fixed non-zero) integers numbers. IN are the natural numbers.
    We regard the function

    F (x,y) := a*x^2 +b*y^2 +c

    Problem :
    We want to find all (x,y) € IN x IN with
    F (x,y) = 0

    Questions:
    1.) Find the conditions for the tripel (a,b,c), so we have:
    A) F (x,y) = 0 has no solution (x,y) € IN x IN
    B) F (x,y) = 0 has exact one solution (x,y) € IN x IN
    C) F (x,y) = 0 has a finite set of solutions (x,y) € IN x IN
    D) F (x,y) = 0 has an infinite set of solutions (x,y) € IN x IN

    [ case C) includes the cases A) and B) ]

    2.) Regard now the case B):
    Is there an algorithm to find this one solution (x,y)?

    3.) Regard further the general case C):
    Is there an algorithm to calculate, how many different solutions (x,y) exist?

    4.) Is the case D) possible with a ≠0, b ≠0, c ≠0 ?!


    Exemples :
    1.) a=1, b=1, c= -6
    x^2 + y^2 - 6 = 0 has no solution (in IN).

    2.) a=13, b= -1, c= -1
    13 x^2 – y^2 - 1 = 0 => (x,y)= (5,18) is the one and only solution.

    3.) a=11, b= -1, c= -7
    11 x^2 – y^2 -7 = 0 has several solutions
    f.e. (1,2) ; (3,13) ; 16, 543) ; (79 , 262), (319, 1058) , ....(???)



    Best regards,
    Winfried
    • 08 Oct 2007, 12:33 am
  • Dr. Maximilian Hasler
    Dr. Maximilian Hasler
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    Re: Diophantic biquadratic function
    Let a,b,c ≠0 three (fixed non-zero) integers numbers. IN are the natural numbers.
    We regard the function
     
    F (x,y) := a*x^2 +b*y^2 +c
     
    Problem : We want to find all (x,y) € IN x IN with F (x,y) = 0
    Roughly, multiplying by "a" the equation is reduced to X²+nY²+C=0 which is of (generalized) Pell type.
    I wrote a somehow detailed answer, but now for the second time my browser had some serious problems and I lost all I wrote before I could post it.
    Quite frustrated, I will now just give some of the links I wanted to suggest:

    http://mathworld.wolfram.com/PellEquation.html
    http://en.wikipedia.org/wiki/Pell's%20equation (esp. link to Lenstra's paper and to indian method)
    http://www.alpertron.com.ar/QUAD.HTM (interactive solver ; "methods" and java source code available)

    I fear that a complete answer to your question would be too long for here.
    But if you found some (partial) answer which can be written without being too complicated, it would be nice if you could post it here.

    Best regards,
    Maximilian
    • 27 Nov 2007, 05:07 am