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prime, prime square, prime cube

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  • Andrey Kulsha
    Hello group, [subj] can form an arithmetic progression, e.g. 23, 25, 27 Are there any more examples known? Best regards, Andrey [Non-text portions of this
    Message 1 of 9 , Feb 23, 2012
      Hello group,

      [subj] can form an arithmetic progression, e.g.

      23, 25, 27

      Are there any more examples known?

      Best regards,

      Andrey

      [Non-text portions of this message have been removed]
    • Jens Kruse Andersen
      ... If 3 numbers in AP is your only condition then there are many small examples. Below are the 42 cases with all numbers below 10^6, sorted by the square and
      Message 2 of 9 , Feb 23, 2012
        Andrey Kulsha wrote:
        > [subj] can form an arithmetic progression, e.g.
        >
        > 23, 25, 27
        >
        > Are there any more examples known?

        If 3 numbers in AP is your only condition then there are many small examples.
        Below are the 42 cases with all numbers below 10^6, sorted by the square
        and then the cube.

        23, 25=5^2, 27=3^3, common difference 2
        541, 1369=37^2, 2197=13^3, common difference 828
        103, 3481=59^2, 6859=19^3, common difference 3378
        5623, 6241=79^2, 6859=19^3, common difference 618
        2467, 16129=127^2, 29791=31^3, common difference 13662
        19507, 24649=157^2, 29791=31^3, common difference 5142
        14869, 32761=181^2, 50653=37^3, common difference 17892
        28549, 39601=199^2, 50653=37^3, common difference 11052
        58831, 69169=263^2, 79507=43^3, common difference 10338
        73951, 76729=277^2, 79507=43^3, common difference 2778
        157, 113569=337^2, 226981=61^3, common difference 113412
        30781, 128881=359^2, 226981=61^3, common difference 98100
        42397, 134689=367^2, 226981=61^3, common difference 92292
        1879, 151321=389^2, 300763=67^3, common difference 149442
        88237, 157609=397^2, 226981=61^3, common difference 69372
        94621, 160801=401^2, 226981=61^3, common difference 66180
        107581, 167281=409^2, 226981=61^3, common difference 59700
        50359, 175561=419^2, 300763=67^3, common difference 125202
        53719, 177241=421^2, 300763=67^3, common difference 123522
        144541, 185761=431^2, 226981=61^3, common difference 41220
        147997, 187489=433^2, 226981=61^3, common difference 39492
        176221, 201601=449^2, 226981=61^3, common difference 25380
        190717, 208849=457^2, 226981=61^3, common difference 18132
        201757, 214369=463^2, 226981=61^3, common difference 12612
        47161, 218089=467^2, 389017=73^3, common difference 170928
        181399, 241081=491^2, 300763=67^3, common difference 59682
        12979, 253009=503^2, 493039=79^3, common difference 240030
        242119, 271441=521^2, 300763=67^3, common difference 29322
        49843, 271441=521^2, 493039=79^3, common difference 221598
        209401, 299209=547^2, 389017=73^3, common difference 89808
        105379, 299209=547^2, 493039=79^3, common difference 193830
        231481, 310249=557^2, 389017=73^3, common difference 78768
        224563, 358801=599^2, 493039=79^3, common difference 134238
        362521, 375769=613^2, 389017=73^3, common difference 13248
        258499, 375769=613^2, 493039=79^3, common difference 117270
        273283, 383161=619^2, 493039=79^3, common difference 109878
        303283, 398161=631^2, 493039=79^3, common difference 94878
        375523, 434281=659^2, 493039=79^3, common difference 58758
        380803, 436921=661^2, 493039=79^3, common difference 56118
        215329, 564001=751^2, 912673=97^3, common difference 348672
        402769, 657721=811^2, 912673=97^3, common difference 254952
        776449, 844561=919^2, 912673=97^3, common difference 68112

        --
        Jens Kruse Andersen
      • Andrey Kulsha
        Thank you Jens for the quick reply. ... Smaller common differences are of greater interest of course :-) Heuristically the number of examples is finite for a
        Message 3 of 9 , Feb 23, 2012
          Thank you Jens for the quick reply.

          > If 3 numbers in AP is your only condition then there are many small
          > examples.
          > Below are the 42 cases with all numbers below 10^6, sorted by the square
          > and then the cube.
          >
          > 23, 25=5^2, 27=3^3, common difference 2
          > 541, 1369=37^2, 2197=13^3, common difference 828
          > 103, 3481=59^2, 6859=19^3, common difference 3378
          > 5623, 6241=79^2, 6859=19^3, common difference 618

          Smaller common differences are of greater interest of course :-)

          Heuristically the number of examples is finite for a given common
          difference.

          Note that the difference can't be +/-1 (mod 5).

          Best regards,

          Andrey
        • WarrenS
          ... THEOREM: The number of examples is finite for a given common difference. BECAUSE: the number of solutions (x,y) of x^2 - y^3 = k is finite for any fixed k,
          Message 4 of 9 , Feb 23, 2012
            > Heuristically the number of examples is finite for a given common difference.

            THEOREM: The number of examples is finite for a given common difference.

            BECAUSE:
            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.
            QED
          • 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 5 of 9 , Feb 24, 2012
              --- 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 6 of 9 , Feb 28, 2012
                >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 7 of 9 , Feb 28, 2012
                  >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 8 of 9 , Feb 28, 2012
                    --- 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 9 of 9 , Mar 4, 2012
                      > 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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