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How the primality of R19 was discovered

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  • julienbenney
    www.pballew.net/arithme4.html The above website contains the answer to a question that has interested me for several years today. I knew R23 was shown to be
    Message 1 of 2 , Apr 30 4:53 AM
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      "www.pballew.net/arithme4.html"

      The above website contains the answer to a question that has
      interested me for several years today. I knew R23 was shown to be
      prime in the late 1920s, but I had no clues when R19 was found to be
      prime.

      However, the site above shows the exact date when the proof of the
      primality of R19 was published - February 14, 1918, just before the
      end of World War I, by an American named Oscar Hoppe.

      Whilst there is no clue as to the exact method by which Hoppe proved
      R19 to be a prime, there is an interesting story about the discovery.
      Before Hoppe's proof, factors were known for all composite repunits up
      to R31 (presumably for R41 and R43 as they have very small factors 83
      and 173 respectively), but no effort to test R19, R23 or (composite)
      R37 for primality had been made. It is odd that the smaller factor of
      R37 had not been found given that Edourd Lucas apparently factored R17
      as early as the 1880s.

      Almost, seemingly, in violation of scientific method, it is recorded
      that Hoppe wrote R19 was prime before beginning his proof! Can you
      imagine doing that today?!
    • djbroadhurst
      ... http://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf Another application of the principle was made by Hoppe in
      Message 2 of 2 , Aug 14, 2013
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        --- In primenumbers@yahoogroups.com,
        "julienbenney" <jpbenney@...> wrote:

        > there is no clue as to the exact method by which Hoppe proved
        > R19 to be a prime

        http://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf

        Another application of the principle was made by Hoppe in
        the investigation of (10^19¬ól)/9. Two proofs of the
        primality of this number were submitted by him to the London
        Mathematical Society. The first proof consisted in isolating
        an unbroken sequence of the 73 smallest prime residues, in
        fact all those <=761. He submitted this proof to Cole, who
        did not consider it sufficient. This seems to indicate that
        Cole had some doubt as to the rigor of his own method. It
        was probably Cole's reply that prompted Hoppe to give an
        independent proof of the primality of his number, as
        described in his second communication to the Society.

        http://plms.oxfordjournals.org/content/s2-17/1/1.1.extract

        leads to

        Thursday, December 6th, 1917.

        Lt.-Col. Cunningham and Dr. Western communicated a paper by
        Mr. 0. Hoppe, "Proof of the Primality of N = (1/9)*(10^19-1)."*

        * A further paper on this subject by Mr. 0. Hoppe was
        communicated on February 14th, 1918; and an Abstract
        containing the results of both papers will be found in the
        Records of that meeting.

        Thursday, February 14th, 1918.

        Lt.-Col. Cunningham and Dr. Western gave an account of a
        further investigation by Mr. 0. Hoppe* on "The Primality of
        (1/9)*(10^19 -1)."

        * See Records of Proceedings, December 6th, 1917.

        ABSTRACTS.

        "The Primality of (1/9)*(10^19-1)."

        Mr. Oscar Hoppe.

        *Mr. Oscar Hoppe's papers deal with the number
        N=(1/9)*(10^19-1) and contain a summary of the results of
        his calculations. He first employed the process for
        factorising large numbers contained in Prof. F. N. Cole's
        paper (Bulletin American Math. Soc., Ser. 2, Vol. 10, p.
        134). He thus proved that p is a quadratic residue of every
        factor of N, where p represents any prime, less than 761, of
        which N is a quadratic residue, those of the form 4n+3
        being taken with a minus sign.

        He then searched for possible prime factors of N up to a
        limit about 23*10^6, as follows. From the properties of such
        factors to moduli 19, 8, 3, 5, 7, 11, 13, every prime factor
        must be congruent to one of 360 residues (mod 2282280). All
        the possible factors of these forms were excluded by using
        the tests of the quadratic character (mods 17, 19, ..., 127).

        Finally, he searched for solutions of N=x^2-y^2, excluding
        possible values of x in a similar manner. The result was that
        no factors were discovered,and that therefore N is a prime.

        He also found that 3^(N-1) = 1 (mod N), which does not
        furnish a proof that N is prime, but confirms the accuracy
        of the other calculations.

        *This Abstract (prepared by Dr. Western) gives a summary of
        the results obtained in two papers communicated at the
        meetings on December 6th, 1917, and February 14th, 1918.

        David
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