--- 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