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Re: [PrimeNumbers] Re: 1993/2011 puzzle

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  • Kevin Acres
    ... Well spotted Mike. 257 really is 2 clues in 1 especially if I mention that 5 also appears in the list - 5, 257, 1399. 1667, 2011. These primes all share
    Message 1 of 15 , Jan 8, 2011
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      At 08:44 AM 9/01/2011, mikeoakes2 wrote:
      >--- In primenumbers@yahoogroups.com, Kevin Acres <research@...> wrote:
      > >
      > > I just checked OEIS and it doesn't give this game away, nor really
      > > offer any help.
      > >
      > > Although I have no historical data for the appropriate year, I should
      > > maybe add that 257 would be a member of the sequence, along with
      > > 1339, 1993 and 2011, if such existed on OEIS.
      >
      >How about this important event:
      >http://www.newadvent.org/fathers/3818.htm
      >
      >Mike

      Well spotted Mike.

      257 really is 2 clues in 1 especially if I mention that 5 also
      appears in the list -> 5, 257, 1399. 1667, 2011. These primes all
      share the same relationship with 1993.

      Which is all probably quite enough to give the game away.

      Kevin.
    • djbroadhurst
      ... I am utterly defeated by this puzzle. Perhaps it is some of retaliation for ... as posted by Andy :-? David
      Message 2 of 15 , Jan 8, 2011
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        --- In primenumbers@yahoogroups.com,
        Kevin Acres <research@...> wrote:

        > 257 really is 2 clues in 1 especially if I mention that 5 also
        > appears in the list -> 5, 257, 1399. 1667, 2011.
        > These primes all share the same relationship with 1993.

        I am utterly defeated by this puzzle.

        Perhaps it is some of retaliation for
        > Australia lost inns & 25 runs v West Indies Perth 30 Jan 1993
        as posted by Andy :-?

        David
      • Kevin Acres
        Hello David, It s not in retaliation, it s just the first puzzle using the prime 2011 that came to mind given the reason behind my latest x=a*x+b search. ...
        Message 3 of 15 , Jan 8, 2011
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          Hello David,

          It's not in retaliation, it's just the first puzzle using the prime
          2011 that came to mind given the reason behind my latest x=a*x+b search.

          At 03:12 PM 9/01/2011, djbroadhurst wrote:
          >--- In primenumbers@yahoogroups.com,
          >Kevin Acres <research@...> wrote:
          >
          > > 257 really is 2 clues in 1 especially if I mention that 5 also
          > > appears in the list -> 5, 257, 1399. 1667, 2011.
          > > These primes all share the same relationship with 1993.
          >
          >I am utterly defeated by this puzzle.
          >
          >Perhaps it is some of retaliation for
          > > Australia lost inns & 25 runs v West Indies Perth 30 Jan 1993
          >as posted by Andy :-?
          >
          >David

          I'm not sure what other hints can I give without totally giving the game away.

          You can be sure that you don't need to think outside the square, as
          the saying goes.

          The entire sequence of these primes, below 10^8, is 2, 5, 257, 1399,
          1667 and 2011. I am also not aware of any others > 10^8. This
          sparseness is both a hint and a feature shared by the generic family.

          Of course, once you see the solution the duality of clues from 257
          will immediately become clear.

          Those who absolutely give up may quickly locate a major clue to
          arriving at the correct solution by googling the name of this group
          followed by 1909 :-)


          Best Regards,

          Kevin (with obviously way too much time on his hands).
        • djbroadhurst
          ... The penny finally dropped :-) Another hint: a quotient too big for the margin. David
          Message 4 of 15 , Jan 9, 2011
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            --- In primenumbers@yahoogroups.com,
            Kevin Acres <research@...> wrote:

            > 257 really is 2 clues in 1 especially if I mention that 5 also
            > appears in the list -> 5, 257, 1399. 1667, 2011.
            > These primes all share the same relationship with 1993.
            ...
            > You can be sure that you don't need to think outside the square,
            > as the saying goes.
            ...
            > Those who absolutely give up may quickly locate a major clue
            > to arriving at the correct solution by googling the name of
            > this group followed by 1909 :-)

            The penny finally dropped :-)

            Another hint: a quotient too big for the margin.

            David
          • djbroadhurst
            ... Puzzle [1993/2001/32933]: Find the next prime in this minimally increasing sequence of primes: 1993, 2011, 32933 ... David
            Message 5 of 15 , Jan 9, 2011
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              --- In primenumbers@yahoogroups.com,
              "djbroadhurst" <d.broadhurst@...> wrote:

              > The penny finally dropped :-)
              > Another hint: a quotient too big for the margin.

              Puzzle [1993/2001/32933]: Find the next prime in this
              minimally increasing sequence of primes: 1993, 2011, 32933 ...

              David
            • djbroadhurst
              ... Kevin has asked me to post my solution. The key to his puzzle was http://en.wikipedia.org/wiki/Fermat_quotient With base a = 1993, the Fermat quotient
              Message 6 of 15 , Jan 9, 2011
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                --- In primenumbers@yahoogroups.com,
                "djbroadhurst" <d.broadhurst@...> wrote:

                > Another hint: a quotient too big for the margin.

                Kevin has asked me to post my solution.
                The key to his puzzle was
                http://en.wikipedia.org/wiki/Fermat_quotient

                With base a = 1993, the Fermat quotient
                (a^(p-1) - 1)/p is divisible by p for
                p = 2, 5, 257, 1399, 1667, 2011
                and for no other prime p < 10^10. In other words,
                2011 is the largest known Wieferich prime in base 1993.

                Another puzzle is to find a sequence of increasing primes
                such that p[n+1] is the smallest Wieferich prime in base p[n]
                for which p[n+1] > p[n].

                Let's call this a "generalized Wieferich sequence".

                For example, the primes
                1153, 1747, 1993, 2011, 32933, 16365127
                form a generalized Wieferich sequence of length 6, with
                16365127 being Kevin's correct solution to my
                "1993/2011/32933 puzzle".

                Puzzle 7: Find a generalized Wieferich sequence of length > 6.

                Comments: This may be solved using primes less than 10^9. So far,
                I have not found a generalized Wieferich sequence of length > 7.

                David
              • djbroadhurst
                ... My interest in generalized Wieferich primes comes from the great achievement of Preda Mihailescu
                Message 7 of 15 , Jan 9, 2011
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                  --- In primenumbers@yahoogroups.com,
                  "djbroadhurst" <d.broadhurst@...> wrote:

                  > Another puzzle is to find a sequence of increasing primes
                  > such that p[n+1] is the smallest Wieferich prime in base p[n]
                  > for which p[n+1] > p[n].
                  >
                  > Let's call this a "generalized Wieferich sequence".
                  >
                  > For example, the primes
                  > 1153, 1747, 1993, 2011, 32933, 16365127
                  > form a generalized Wieferich sequence of length 6, with
                  > 16365127 being Kevin's correct solution to my
                  > "1993/2011/32933 puzzle".
                  >
                  > Puzzle 7: Find a generalized Wieferich sequence of length > 6.

                  My interest in generalized Wieferich primes comes from the great
                  achievement of Preda Mihailescu
                  http://en.wikipedia.org/wiki/File:450px-Preda_Mihailescu_vor_Tafel.png
                  one of the authors of OpenPFGW, who proved the Catalan Conjecture:
                  http://www.ams.org/journals/bull/2004-41-01/S0273-0979-03-00993-5/S0273-0979-03-00993-5.pdf

                  Note that the reviewer says that Preda was

                  "a mathematician practically unknown to the experts in this area"

                  who

                  "turned up with a complete proof of the conjecture."

                  Moreover,

                  "his proof has very little to do with computation, making instead
                  use of deep theoretical results, notably from the theory of
                  cyclotomic fields. Mihailescu, born in 1955 in Romania, received
                  his mathematical education at the ETH Zurich. He has worked in
                  the machine and finance industry but is now doing research in
                  Germany at the University of Paderborn. The present article
                  describes briefly the landmarks in the history of the work on
                  Catalan's problem and outlines Mihailescu's brilliant solution."

                  So next time that someone tells you that you are
                  "unknown to the experts", remember Preda :-)

                  David
                • Makoto Kamada
                  ... 197, 653, 1381, 1777, 6211, 39041, 144449603 Makoto Kamada
                  Message 8 of 15 , Jan 9, 2011
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                    > Puzzle 7: Find a generalized Wieferich sequence of length> 6.

                    197, 653, 1381, 1777, 6211, 39041, 144449603

                    Makoto Kamada
                  • djbroadhurst
                    ... Congratulations to Makoto for a valid solution, with the last 2 primes recorded in http://www.cecm.sfu.ca/~mjm/WieferichBarker/Data/q5p9.txt In addition to
                    Message 9 of 15 , Jan 9, 2011
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                      --- In primenumbers@yahoogroups.com,
                      Makoto Kamada <m_kamada@...> wrote:

                      >> Puzzle 7: Find a generalized Wieferich sequence of length > 6.
                      > 197, 653, 1381, 1777, 6211, 39041, 144449603

                      Congratulations to Makoto for a valid solution,
                      with the last 2 primes recorded in
                      http://www.cecm.sfu.ca/~mjm/WieferichBarker/Data/q5p9.txt

                      In addition to Makoto's solution, I found the sequence
                      19739, 61729, 445631, 508009, 728437, 1051139, 5176948723

                      Now comes a harder puzzle, which I have not solved :-(

                      > Another puzzle is to find a sequence of increasing primes
                      > such that p[n+1] is the smallest Wieferich prime in base p[n]
                      > for which p[n+1] > p[n].
                      > Let's call this a "generalized Wieferich sequence".

                      Puzzle 8: Find a generalized Wieferich sequence of length > 7.

                      David
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