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Proof of less-fortunate numbers

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  • John W. Nicholson
    Proof of less-fortunate numbers (as seen on http://primes.utm.edu/glossary/page.php?sort=FortunateNumber ): By observation: The largest prime of the solution
    Message 1 of 1 , Feb 20, 2005
      Proof of less-fortunate numbers (as seen on
      http://primes.utm.edu/glossary/page.php?sort=FortunateNumber ):

      By observation: The largest prime of the solution for each
      primordial to "Carl Pomerance
      Observation" by Said El Aidi on
      http://www.primepuzzles.net/problems/prob_002.htm are less-fortunate
      numbers.

      Knowing the above fact, how big does the size of the gap to the next
      prime, assuming with and without that p_n# (+/-) 1 is prime, have to
      be in order to fail to be a Fortunate Number? Chris stated on the
      page (I changed the notation) "So we are looking for a prime gap
      near p_n# of about (log p_n# log log p_n#)^2." But two things are
      for sure, there is a gap with a size >= p_n after p_n# and . (For
      the same reason that there is a gap after p_n! >= p_n but with a
      common multiple added to p_n#.) Is this the only thing that can be
      said about this gap?

      Can a modification of Aidi's work be stated to prove a Fortunate
      number, q, exist between p_n# + p_n < q < p_n# + (p_(n+1))^2, so
      that r = q - p_n# with r being prime? Does knowing the status of
      less-fortunate numbers satisfy a necessary or sufficient condition
      for Fortunate number q? I am thinking that the number of primes in
      the Aidi range is greater than this range but there is always one
      prime because of the required solutions to the congruence of numbers
      between p_n# and p_n! in the range p_n# + p_n < q < p_n# + (p_(n+1))
      ^2. Another words the solutions must be spread out, this small group
      can not be the last grouping.

      On a different note, I am rewriting Aidi's statements, I am hoping
      someone can read and send suggestions back to me about the
      document. The two page paper is in MSWord format and is not
      downloadable to the web page. Can someone look at it by writing me
      personally?

      Thanks for your answers,

      John
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