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23298Re: A question

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  • Dimiter Skordev
    Oct 1, 2011
      --- In primenumbers@yahoogroups.com, "mikeoakes2" <mikeoakes2@...> wrote:
      > > I can't think of a reason why any prime would be excluded from being a factor of numbers in the set.
      > I agree.
      After the work Mark and Mike did, it really seems that any prime number is Euclid-style accessible. Unfortunately, I have no idea how this could be proved.

      > > I've only gone 5 'deep', and thus far 19 is the only
      > > prime < 29 that hasn't shown up yet. 11 and 17 were
      > > late to the party, but they eventually surfaced at depth 5.
      > A is defined recursively so, rather than your "depth" (which I'm not sure I understand) I favour a measure "recursion_depth", which would enumerate the elements of A in order of their appearance as follows:-
      > x*y recursion_depth
      > 1 1
      > 1*2 2
      > 2*3 3
      > 6*7 4
      > 42*43 5
      > 1806*13 6
      > 1806*139 6
      > etc.
      I also would understand the notion of depth in this way. When I wrote "No, I have not gone even to that depth", I actually meant I did not go to a depth at which, for instance, 11 and 17 are already shown to be Euclid-style accessible.

      In addition to this notion of accessibility, maybe a stronger one also deserves some attention, namely the one we get when we replace "y is a prime divisor of x+1" with "y is the least prime divisor of x+1" in the definition of the set A.

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