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Re: prime, prime square, prime cube

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  • djbroadhurst
    ... The actual history is both more complex and more interesting than Warren s mistaken attribution. J.W.S. Cassels, Mordell s finite basis theorem revisited,
    Message 1 of 9 , Feb 24, 2012
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      --- In primenumbers@yahoogroups.com,
      "WarrenS" <warren.wds@...> wrote:

      > the number of solutions (x,y) of x^2 - y^3 = k
      > is finite for any fixed k,
      > proved by LJ Mordell sometime between 1905-1925.

      The actual history is both more complex and more
      interesting than Warren's mistaken attribution.

      J.W.S. Cassels,
      Mordell's finite basis theorem revisited,
      Math. Proc. Camb. Phil. Soc. 100 (1986) 31-40

      points out that in the conclusion of

      L. J. Mordell,
      The diophantine equation y^2 — k = x^3,
      Proc. London Math. Soc. (2) 13 (1913) 60-80,

      Mordell (not knowing of Thue's work) and was led,
      for unknown reasons, to make a (false) claim directly
      opposed to the result that Warren has attributed to him.

      I looked to see if that is the case. Indeed it is.
      Mordell concluded:

      >> When k = 1, 4, 6, 7, 11, 18, 14, 16, 20, 21, 28, 25, 27,
      29, 82, 34, 89, 42, 45, 46, 47, 49, 51, 58, 58, 59, 60, 61,
      62, 66, 67, 69, 70, 75, 77, 78, 88, 84, 85, 86, 87, 88, 90,
      98, 95, 96, the equations are insoluble or admit only a
      limited number of solutions. For the remaining values of k,
      there are an infinite number of solutions, except when
      k = 74, in which case nothing can be said about the equation.<<

      Mordell's 1913 paper gained him the Smith Prize,
      but failed to gain him a Fellowship of St John's.

      It seems that Mordell came to realize his error after studying

      E. Landau and A. Ostrowski,
      On the diophantine equation a*y^2 + b*y + c = d*x^n,
      Proc. London Math. Soc. (2) 19 (1920) 276-280

      and commented (in 1922) that this immediately
      implies that he was wrong in 1913.

      However, it appears that the paper by Landau and Ostrowski
      appeared after the same result had been published in

      A. Thue, Über die Unlösbarkeit der Gleichung
      a*x^2 + b*x + c = d*y^n in grossen ganzen Zahlen,
      Arch. math, og naturv. Kristiania 34 (1917) no. 16.

      Of the 4 papers that I cite above,
      I have read only the first two.

      David
    • WarrenS
      ... --David, thanks a lot for this help. I in fact was going to look into this myself but unfortunately my library only allowed me to access one of Mordell s 3
      Message 2 of 9 , Feb 28, 2012
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        >DJ Broadhurst:
        > The actual history is both more complex and more
        > interesting than Warren's mistaken attribution.

        --David, thanks a lot for this help.
        I in fact was going to look into this myself but unfortunately my library only allowed
        me to access one of Mordell's 3 papers, namely the one you mentioned
        PLMS 13 (1913) 60-80 which as you said (& I already knew) was not doing the job.

        But I have reason to believe, but unconfirmed since I do not have access to these papers,
        that one or both of the following papers by Mordell do do the job:

        Cambridge Philos Soc Proc 21(1922) 179-192.
        Quart J Pure Appl Math 45 (1914) 170-186.
      • WarrenS
        ... --David, thanks a lot for this help. I in fact was going to look into this myself but unfortunately my library only allowed me to access one of Mordell s 3
        Message 3 of 9 , Feb 28, 2012
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          >DJ Broadhurst:
          > The actual history is both more complex and more
          > interesting than Warren's mistaken attribution.

          --David, thanks a lot for this help.
          I in fact was going to look into this myself but unfortunately my library only allowed
          me to access one of Mordell's 3 papers, namely the one you mentioned
          PLMS 13 (1913) 60-80 which as you said (& I already knew) was not doing the job.

          But I have reason to believe, but unconfirmed since I do not have access to these papers,
          that one or both of the following papers by Mordell do do the job:

          Cambridge Philos Soc Proc 21(1922) 179-192.
          Quart J Pure Appl Math 45 (1914) 170-186.
        • djbroadhurst
          ... I not have read either of those. I have, however, read L. J. Mordell, On the Integer Solutions of the Equation, e*y^2 = a*x^3+b*x^2+c*x+d, Proc. London
          Message 4 of 9 , Feb 28, 2012
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            --- In primenumbers@yahoogroups.com,
            "WarrenS" <warren.wds@...> wrote:

            > > The actual history is both more complex and more
            > > interesting than Warren's mistaken attribution.
            >
            > I have reason to believe, but unconfirmed since I do not
            > have access to these papers, that one or both of the
            > following papers by Mordell do do the job:
            > Cambridge Philos Soc Proc 21(1922) 179-192.
            > Quart J Pure Appl Math 45 (1914) 170-186.

            I not have read either of those. I have, however, read

            L. J. Mordell,
            On the Integer Solutions of the Equation,
            e*y^2 = a*x^3+b*x^2+c*x+d,
            Proc. London Math. Soc. s2-21(1) (1923) 415-419
            doi:10.1112/plms/s2-21.1.415

            where Mordell concedes that this problem
            was solved by Thue in 1909, with a result very
            different from Mordell's false claim of 1913.

            In Cassel's obituary of Mordell:

            Louis Joel Mordell, 1888-1972,
            by J. W. S. Cassels, in
            Biographical Memoirs of Fellows of the Royal Society,
            Vol. 19 (Dec., 1973), pp. 493-520.

            there is mention of an earlier retraction, circa 1919-1920:

            >> In a paper which was little noticed at the time Thue
            had already shown that the equation (2.7) has only
            finitely many integral solutions for any cubic form
            (or more generally, for any other than a power of a
            binary quadratic). If Mordell had known of this theorem,
            he would have deduced at once that y^2 = x^3+ k has
            only finitely many integral solutions x,y; at least with
            x prime to 2k. As a matter of fact he only learned
            of Thue's result later (paper 15) and at the time
            he believed that there could be infinitely many solutions
            for some k (cf. end of paper 2).<<

            "Paper 15" seems to be:

            L J Mordell,
            A statement by Fermat,
            Proc. Lond. Math. Soc., (2) 18 (1920) v.

            The index for that volume indicates that this paper
            was read in 1919, but not printed in the journal.
            So 1919 is the earliest date for a retraction, by Mordell,
            of his false claim of 1913, that I seen cited.

            David
          • Kermit Rose
            ... Unless there just happens to be one or more of the form 5, x^2 = 5 + k, y^3 = 5 + 2k Kermit
            Message 5 of 9 , Mar 4 5:03 AM
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              > Note that the difference can't be ±1 (mod 5).

              > Best regards,

              > Andrey


              Unless there just happens to be one or more of the form

              5, x^2 = 5 + k, y^3 = 5 + 2k

              Kermit
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