## Re: A question

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• ... 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
Message 1 of 12 , Oct 1, 2011
--- 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
• ... PRP27 = 643679794963466223081509857 PRP28 = 2496022367830647867616317307 PRP44 = 20316223246552213835636779619145529457704309 [Non-text portions of this
Message 2 of 12 , Oct 1, 2011
>
> 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).
>

PRP27 = 643679794963466223081509857
PRP28 = 2496022367830647867616317307
PRP44 = 20316223246552213835636779619145529457704309

[Non-text portions of this message have been removed]
• ... Thanks, Bernardo. Now do you feel like factorising the 98+27=125-digit integer (u[29]+1) ... Mike
Message 3 of 12 , Oct 1, 2011
--- In primenumbers@yahoogroups.com, Bernardo Boncompagni <RedGolpe@...> wrote:
>
> >
> > 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).
> >
>
> PRP27 = 643679794963466223081509857
> PRP28 = 2496022367830647867616317307
> PRP44 = 20316223246552213835636779619145529457704309
>

Thanks, Bernardo.

Now do you feel like factorising the 98+27=125-digit integer (u[29]+1)
:-)

Mike
• ... Somebody has been here before, carried it up to 256 digits, and saved it all in the factordb http://factorization.ath.cx/index.php?id=1100000000024656542
Message 4 of 12 , Oct 1, 2011
• ... We only need the smallest factor which is trivially found by trial factoring to be 103. The sequence of smallest prime factors is the Euclid-Mullin
Message 5 of 12 , Oct 1, 2011
Mike wrote:
> Now do you feel like factorising the 98+27=125-digit integer (u[29]+1)

We only need the smallest factor which is trivially found by trial
factoring to be 103.
The sequence of smallest prime factors is the Euclid-Mullin sequence.
See for example http://oeis.org/A000945 and
http://en.wikipedia.org/wiki/Euclid%E2%80%93Mullin_sequence

The smallest missing prime in the 47 known terms is 31.

--
Jens Kruse Andersen
• ... It is notable that only one known factorization is incomplete: http://www.rieselprime.de/Others/EuclidMullin.htm and in that case it suffices to show that
Message 6 of 12 , Oct 1, 2011