--- In

primenumbers@yahoogroups.com, "Dimiter Skordev" <skordev@...> wrote:

>

> 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.

>

In this case, of course, the tree degenerates into a list, and so is more straightforward to program; and we have simply

card(A)=recursion_depth.

Defining the function s(m), for integer m, to be its smallest prime factor, your set A is the set {u[n]}, with the (monotonically increasing) sequence u[] being defined by:

u[1] = 1

u[n] = u[n-1] * s(u[n-1]+1), n >= 2

As with the earlier definition, I can see no reason why all primes should not be accessible.

I removed any restriction on size of integer to (attempt to) factorise.

For 14 <= recursion_depth <= 28, the first inaccessible prime is 19.

I have now run into a brick wall:

u[28]=326408399720836161014262419152749\

231088487789442706259494316079679210912\

54869713984092316692981590

and pari-Gp is struggling to factorise the 98-digit integer (u[28]+1).

ecm anyone?

Mike