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Re: sqrt(Phi(n,b)) for n>2 and b>1

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  • djbroadhurst
    ... There was an error in Part (6), giving an upper bound for the probability of a solution with n 100, sorry. ... [here I should have computed:]
    Message 1 of 55 , Dec 22, 2010
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      --- In primenumbers@yahoogroups.com,
      "djbroadhurst" <d.broadhurst@...> wrote:

      > This is the best that I could do, by way of an heuristic argument
      > that the probability of another (n,b) pair is less than 1/10^6.

      There was an error in Part (6), giving an upper bound
      for the probability of a solution with n > 100, sorry.

      Here I correct it, with no change to the conclusion:

      > 6) For n > 6 we know from
      > http://mathworld.wolfram.com/TotientFunction.html
      > that eulerphi(n) > sqrt(n).
      > Hence I estimate the probability of a solution
      > with n > 100 as less than

      [here I should have computed:]

      {print(precision(sum(n=10^2,10^4,1./4^(eulerphi(n)/2))
      +suminf(n=10^4,1/2^(sqrt(n)/2)),1));}

      0.0000000005135754773441919258

      [with the same conclusion:]

      > which is far less than the estimate 2/10^7, in Part (4).
      >
      > 7) In conclusion, I believe that there is a probability
      > of less than 1/10^6 for finding a second integer pair (n,b),
      > with n>2 and b>1, such that sqrt(Phi(n,b)) is an integer.

      David
    • djbroadhurst
      We have written up (some of) our findings: http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752 Kevin and David
      Message 55 of 55 , Feb 11, 2011
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        We have written up (some of) our findings:

        http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind1102&L=nmbrthry&P=R752

        Kevin and David
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