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23295Re: A question

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  • mikeoakes2
    Oct 1, 2011
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      --- In primenumbers@yahoogroups.com, "Mark" <mark.underwood@...> wrote:
      >
      > --- In primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@> wrote:
      > >
      > > Let A be the least set of natural numbers with the following two properties:
      > > (i) the number 1 belongs to A;
      > > (ii) whenever x belongs to A, and y is a prime divisor of x+1, the product x*y also belongs to A.
      > > Let us call a prime number Euclid-style accessible if it is a divisor of some number belonging to A. Are there prime numbers that are not Euclid-style accessible, and if there are such ones, which is the least among them?
      > >
      >
      > I can't think of a reason why any prime would be excluded from being a factor of numbers in the set.

      I agree.

      > 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.

      Going to recursion_depth=13, pari_GP take a couple of minutes to find that all primes < 271 are Euclid-style accessible.

      2621 elements have been inserted into A at this recursion_depth; but 86 others have not been inserted, since they exceed the size limit of 10^64 which I imposed on numbers so that they can be factorised in at most a few seconds.

      Mike
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