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Re: Yet another factoring puzzle

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  • djbroadhurst
    ... We seek complete factorization of both of the cofactors 25^k-4*5^k-1 and 25^k+4*5^k-1 for some k 300, where each has more than 400 decimal digits. We had
    Message 1 of 23 , Sep 19, 2013
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      Jean-Louis Carton wrote:

      > So for both factors to be prime the probability is about 0.000001.

      We seek complete factorization of both of the cofactors
      25^k-4*5^k-1 and 25^k+4*5^k-1
      for some k > 300, where each has more than
      400 decimal digits. We had better avoid
      the case k = 0 mod 3, where each cofactor
      has an algebraic factorization, which is
      here a distinct disadvantage.

      Suppose that we sieve out primes to depth d
      and hope for what is left to yield a
      a pair of PRPs as here, with k = 304:

      (25^304-4*5^304-1)/(2^2*11*29*1289*1759*9511*27851) is 3-PRP!
      (25^304+4*5^304-1)/(2^2*1439*17390951) is 3-PRP!

      The probability for success for a single value of k
      coprime to 3 is of order

      (exp(Euler)*log(p)/log(25))^2/k^2

      Setting p = 2*10^7 and summing over /all/ k > 300,
      the probability that Exercise 6 has /no/ solution is

      exp(-2/3*(exp(Euler)*log(2*10^7)/log(25))^2/301) =~ 83%

      In fact it has a solution almost immediately, at k = 304.

      David
    • Phil Carmody
      ... But is that more or less remarkable than the expectation of any one of Phil Taylor s darts landing in the region around where it actually landed? You only
      Message 2 of 23 , Sep 25, 2013
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        On Mon, 9/16/13, djbroadhurst wrote:
        --- "djbroadhurst" <d.broadhurst@...> wrote:
        > > Exercise 6: Find the complete factorization of F(n) for at
        > > least one even integer n > 600.
        > As far as I can tell, no-one (apart from the setter)
        > yet solved Exercise 6, which can be done in less
        > than 2 minutes, using OpenPFGW. What is remarkable
        > about this exercise is that it can be solved so
        > quickly. Heuristically, that was not to be expected.

        But is that more or less remarkable than the expectation of
        any one of Phil Taylor's darts landing in the region around
        where it actually landed? You only chose that target after
        the arrow had landed, I'm sure.

        How many mathematical diversions have you looked at
        via the medium of numerical computation? How many of
        them would you expect to be remarkably easier than
        expected to solve? Probably a non-zero answer. Don't
        be surprised that one particular example was one.

        Knowing what he's trying to say, even if he's not getting
        it across clearly,
        Phil
        --
        () ASCII ribbon campaign () Hopeless ribbon campaign
        /\ against HTML mail /\ against gratuitous bloodshed

        [stolen with permission from Daniel B. Cristofani]
      • djbroadhurst
        ... It happened thus: 1) I determined to factorize F(n)=((n^2-9)/4)^2-5 for n
        Message 3 of 23 , Sep 27, 2013
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           ---In primenumbers@yahoogroups.com, <thefatphil@...> wrote:

           

          > You only chose that target after
          > the arrow had landed, I'm sure.

           

          It happened thus:

           

          1) I determined to factorize F(n)=((n^2-9)/4)^2-5 for
          n <= 300, completely. As later shown in "factordb", I succeeded.

           

          2) Meanwhile I ran OpenPFGW on n in [301,600], hoping for a
          quick outlier and found none.

           

          3) I estimated the probability of an easily discoverable
          completely factorization for n>600 and found it to be small.

           

          4) Recalling how I had once been caught out before by
          a "probably no more" heuristic, I set a lone process running on
          n in [601, 10000]  so as not to be caught out again by Jens.

           

          5) When I later looked  and pfgw.log, it had found a hit at
          n=608.

           

          So yes, Phil, you are quite correct that the puzzle was set
          after this finding. However the heuristic that I gave was
          made prior to my discovery, else I would not have said that
          I was surprised.

           

          The point that you are making (I think) is that I do such 
          expsriments often and only notice when the result is unexpected.
          I don't tell folk about all the boring times when a negative
          heuristic is borne out by a null result. That is the selection

          effect.

           

          David (guilty of not boring folk with what is routine)

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