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22784Re: Old conjecture of Euler about Zeta(3) almost certainly wrong

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  • djbroadhurst
    Jul 2, 2011
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > --- In primenumbers@yahoogroups.com,
      > "WarrenS" <warren.wds@> wrote:
      >
      > > Leonhard Euler [Opera Omnia Ser.1 vol.4 pp.143-144]
      > > conjectured that there was an integer linear relation amongst
      > > [ Zeta(3), ln(2)^2, ln(2)*Pi^2 ].
      >
      > No. The 3 terms were [Zeta(3), ln(2)^3, ln(2)*Pi^2]
      > and far from making a conjecture, Euler arrived at the
      > conclusion that no integer relation exists.
      >
      > A little scholarship revealed the following translation
      > of what Euler actually said, in the conclusion of
      > "De relatione inter ternas pluresve quantitates instituenda",
      > presented to the St. Petersburg Academy, 14 August 1775:
      >
      > "It would be superfluous to continue these operations further,
      > since one may now understand sufficiently well from here,
      > that no relationship between the three quantities taken is
      > given that is able to be resolved consistent with the truth.
      > Therefore, since I tried in vain to explore the investigation
      > of this reciprocal sum of cubes in different ways and this
      > method was called into use without result, it seems from
      > investigation that it rightly must be abandoned."
      >
      > http://eulerarchive.maa.org/docs/translations/E591trans.pdf

      Off list, Warren told me that both the typo
      (wrong power of log(2)) and also the error of judgment
      as to what Euler concluded came from this popular account:

      http://tinyurl.com/3eqzfdo

      David
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