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13551[PrimeNumbers] Re: 2*a^a+-1 primes

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  • mikeoakes2@aol.com
    Sep 16, 2003
      As Harsh announced here a while back, there is a small coordinated search
      going on for primes of this form - see:
      http://groups.yahoo.com/group/primenumbers/files/2%2Aa%5Ea%2B-1%20prime%20sear
      ch/index.htm

      so it is instructive to see how much more or less likely such numbers are to
      be prime than "random" integers of the same size.

      Here are the results of removing all prime factors < p_max for both the "-"
      and "+" forms, compared with the same exercise for a set of numbers of the form
      2*a^a+r, where the "r" values are selected randomly from the range 1<=r<=10^9.

      A range of 20000 <= a <= 25000 was used, for all 3 cases, and the columns in
      the following table record how many of the 5000 starting numbers remain after
      sieving to the depth p_max:-

      p_max "-" form "+" form "random" form
      2*10 2734 1650 1698
      55 2286 1367 1365
      2*10^2 1856 1167 1198
      2*10^3 1395 861 829
      2*10^4 1020 675 642
      2*10^5 592 318 479
      2*10^6 501 247 420
      2*10^7 431 205 ??
      2*10^8 381 178 ??
      2*10^9 351 156 ??

      Phil Carmody's excellent "aagenm.exe" was used to obtain the results in the
      2nd and 3rd columns, in about 0.8GHz-days.
      For the final column PFGW was used, but its trial division is about 100 times
      slower than Phil's program's, and so would have taken months to find all the
      ?? figures.

      It is clear that more primes are to be expected from the "-" form than either
      from the "+" form or from a collection of "ordinary" (odd) numbers of the
      same size.

      The p_max=55 row is included to enable a comparison to be made with the
      theoretical prediction from my earlier posts giving the probabilities of the "-"
      and "+" forms being divisible by each of the primes <= 53:-

      p_max "-" form "+" form "random" form
      55 2221.8 1419.8 1360.9

      where the last column is of course just 5000*(1-1/3)*(1-1/5)*...*(1-1/53).

      Mike


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