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On the Diophantine equations

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  • Andrey Kulsha
    Is it true that for every given set of integers (p,q,r) with min(p,q,r) 1 and max(p,q,r) 2 equation of the form a^p + b^q = c^r has no more than finite number
    Message 1 of 3 , Feb 20, 2008
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      Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form

      a^p + b^q = c^r

      has no more than finite number of solutions in positive integers (a, b, c)?

      Regards,

      Andrey

      [Non-text portions of this message have been removed]
    • Andrey Kulsha
      ... I mean distinct positive integers (a, b, c). Indeed, e.g., (n^2-1)^3 + (n^2-1)^2 = (n^3-n)^2 form an infinite number of solutions for (p,q,r) = (3,2,2).
      Message 2 of 3 , Feb 20, 2008
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        > Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form
        >
        > a^p + b^q = c^r
        >
        > has no more than finite number of solutions in positive integers (a, b, c)?

        I mean distinct positive integers (a, b, c).

        Indeed, e.g., (n^2-1)^3 + (n^2-1)^2 = (n^3-n)^2 form an infinite number of solutions for (p,q,r) = (3,2,2).

        Best regards,

        Andrey

        [Non-text portions of this message have been removed]
      • Jack Brennen
        ... See the Fermat-Catalan conjecture. http://mathworld.wolfram.com/Fermat-CatalanConjecture.html
        Message 3 of 3 , Feb 21, 2008
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          Andrey Kulsha wrote:
          > Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form
          >
          > a^p + b^q = c^r
          >
          > has no more than finite number of solutions in positive integers (a, b, c)?
          >


          See the Fermat-Catalan conjecture.

          http://mathworld.wolfram.com/Fermat-CatalanConjecture.html
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