--- 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.

Dimiter