[seqfan] Re: 13532385396179 precursors

[Robert Gerbicz]

If the numbers are really random, then the probability is (roughly)
1/(k*log(10)).

On the next (second) layer there are still primes since 13^532385396179
begins
with 689710669368417039950761288661178934104483195797695497357742398030694745295986525
and the first few digits of this is prime:
68971066936841703995076128866117893410448319579 and
68971066936841703995076128866117893410448319579769
using more digits we can get two more primes: at the first 198 digits and
at 449 digits, and there is no more in the first 2000 digits.
If we take these numbers then on the next (third) layer there could be more
than a trillion of numbers.

2017-06-10 5:52 GMT+02:00 Neil Sloane <njasloane at gmail.com>:

> Neat!  So there are surely infinitely many counterexamples
> (even though there is no proof that a prime prefix exists,
> it should exist on probabilistic grounds - take your Huge number H =
> 13^532385396179,
> take the first k digits,
> that is a prime with prob 1/k, and Sum 1/k diverges)
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Fri, Jun 9, 2017 at 11:42 PM, Hans Havermann <gladhobo at bell.net> wrote:
>
> > > Will this process eventually end? Are the total number of precursors,
> > precursors of precursors, precursors of precursors of precursors, etc.
> > finite?
> >
> > Think about the number 13^532385396179. Take some of its initial decimal
> > digits that make those digits prime and raise this to the power of the
> rest
> > of the digits. Now you have a new (larger) number. Repeat.
> >
> > --
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> >
>
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