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How could one show without computers that there was no number x^2-y^2 that equalled R19

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  • julienbenney
    I was looking at Oscar Hoppe’s proof of the primality of 1,111,111,111,111,111,111 (R19) and it apparently involved, according to
    Message 1 of 6 , May 29, 2014
      I was looking at Oscar Hoppe’s proof of the primality of 1,111,111,111,111,111,111 (R19) and it apparently involved, according to ‘http://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf’ a proof that R19 was not the difference of any pair of squares.

      Whilst I do have a definite idea how that would be established, I do not know how it could be done without computers (that was the World War I period, after all) for a number the size of R19??
    • Jose Ramón Brox
      Hi jp! I don t know if I am understanding you well. The only natural numbers that cannot be written as a difference of two squares are those of the form 4n+2,
      Message 2 of 6 , May 29, 2014
        Hi jp!

        I don't know if I am understanding you well. The only natural numbers that cannot be written as a difference of two squares are those of the form 4n+2, i.e., the numbers which are even but not divisible by 4; the rest of them are all differences of two squares. In particular, if n=2k+1 is an odd number, then (k+1)^2-k^2 = 2k+1 = n.

        Therefore R19 IS a difference of two squares.

        I cannot see the claim you mentioned in the paper you cited. Can you give the page?

        Regards,
        Jose Brox




        2014-05-29 15:56 GMT+02:00 jpbenney@... [primenumbers] <primenumbers@yahoogroups.com>:
         

        I was looking at Oscar Hoppe’s proof of the primality of 1,111,111,111,111,111,111 (R19) and it apparently involved, according to ‘http://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf’ a proof that R19 was not the difference of any pair of squares.

        Whilst I do have a definite idea how that would be established, I do not know how it could be done without computers (that was the World War I period, after all) for a number the size of R19??




        --
        La verdad (blog de raciocinio político e información social)
      • Jane Sullivan
        On 29/05/2014 20:28, Jose Ramón Brox ambroxius@gmail.com [primenumbers] ... Shouldn t it say something about being the difference of two squares in two
        Message 3 of 6 , May 29, 2014
          On 29/05/2014 20:28, Jose Ramón Brox ambroxius@... [primenumbers]
          wrote:
          > Hi jp!
          >
          > I don't know if I am understanding you well. The only natural numbers
          > that cannot be written as a difference of two squares are those of the
          > form 4n+2, i.e., the numbers which are even but not divisible by 4; the
          > rest of them are all differences of two squares. In particular, if
          > n=2k+1 is an odd number, then (k+1)^2-k^2 = 2k+1 = n.
          >
          > Therefore R19 IS a difference of two squares.
          >
          > I cannot see the claim you mentioned in the paper you cited. Can you
          > give the page?

          Shouldn't it say something about being the difference of two squares in
          two different ways? If a number N = a^2 - b^2 = (a+b)(a-b) but a-b = 1
          you don't get a factorization, and N will be a+b which could be prime;
          but if N is also c^2 - d^2, where none of a, b, c, and d equals any of
          the others, then you will get a factorization.

          >
          > Regards,
          > Jose Brox
          >
          >
          >
          >
          > 2014-05-29 15:56 GMT+02:00 jpbenney@... <mailto:jpbenney@...>
          > [primenumbers] <primenumbers@yahoogroups.com
          > <mailto:primenumbers@yahoogroups.com>>:
          >
          > __
          >
          > I was looking at Oscar Hoppe’s proof of the primality of
          > 1,111,111,111,111,111,111 (R19) and it apparently involved,
          > according to
          > ‘http://www.ams.org/journals/bull/1930-36-12/S0002-9904-1930-05077-6/S0002-9904-1930-05077-6.pdf%e2%80%99
          > a proof that R19 was not the difference of any pair of squares.
          >
          > Whilst I do have a definite idea how that would be established, I do
          > not know how it could be done without computers (that was the World
          > War I period, after all) for a number the size of R19??
          >
          >
          >
          >
          > --
          > La verdad (blog de raciocinio político e información social)
          > <http://josebrox.blogspot.com/>
          >
          >


          Best wishes
          --
          Jane Sullivan
          Beckenham
        • djbroadhurst
          I already tried to help with Julien this topic, ... Thursday, December 6th, 1917. Lt.-Col. Cunningham and Dr. Western communicated a paper by Mr. 0. Hoppe,
          Message 4 of 6 , May 30, 2014
            I already tried to help with Julien this topic,
            by consulting the records of the LMS:

            ----------

            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.

            -----

            What more might one ascertain at this distance of time?

            David
          • djbroadhurst
            On page 160 of http://archive.org/details/thoriedesnombres01krai Kraitchik completes the factorization of 2^61 + 2^31 + 1 = 3456749*667055378149 after 9 pages
            Message 5 of 6 , Jun 1, 2014
              On page 160 of
              http://archive.org/details/thoriedesnombres01krai
              Kraitchik completes the factorization of
              2^61 + 2^31 + 1 = 3456749*667055378149
              after 9 pages of detailed consideration of
              N = a^2 - b^2.
              He then asserts that the same method factorizes
              2^53 + 2^27 + 1 = 15358129*586477649
              and proves the primality of (10^19-1)/9.

              David
            • djbroadhurst
              PS: Kraitchik failed to achieve the factorization of Phi(110,3) = 16143694150072550161 = 659671*24472341743191 ... David ... On page 160 of
              Message 6 of 6 , Jun 1, 2014
                PS: Kraitchik failed to achieve the factorization of
                Phi(110,3) = 16143694150072550161 = 659671*24472341743191
                and claimed a "degree of moral certitude" that this number is prime:
                 
                > ... la recherche de ce facteur aurait été une perte de temps.
                > 12. Reste à apprécier le degré de certitude morale qu'on a
                > pour affirmer que N est premier.

                David

                ---In primenumbers@yahoogroups.com, <david.broadhurst@...> wrote :

                On page 160 of
                http://archive.org/details/thoriedesnombres01krai
                Kraitchik completes the factorization of
                2^61 + 2^31 + 1 = 3456749*667055378149
                after 9 pages of detailed consideration of
                N = a^2 - b^2.
                He then asserts that the same method factorizes
                2^53 + 2^27 + 1 = 15358129*586477649
                and proves the primality of (10^19-1)/9.

                David
                 
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