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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: Re: Another Prime Number Question With No Practical Application
Message-ID: <298@faron.UUCP>
Date: Fri, 10-May-85 08:43:33 EDT
Article-I.D.: faron.298
Posted: Fri May 10 08:43:33 1985
Date-Received: Sat, 11-May-85 08:49:04 EDT
References: <226@ihnet.UUCP> <574@lll-crg.ARPA>
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Organization: The MITRE Coporation, Bedford, MA
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> In article <226@ihnet.UUCP> eklhad@ihnet.UUCP (K. A. Dahlke) writes:
> >< and other diversionary tactics >
> >	How many primes, when written in base 10,
> >also produce prime sub-numbers (looking at the first n digits)?
> >For example: 7193 is in the set, since
> >7, 71, 719, and 7193 are all prime.
> >The list begins: 3, 5, 7, 31, 37, 53, 59, 71, 73, 79, 
> >311, 313, 317, 373, 379, 593, 599, ...
> >Is the list infinite?
> >If so, can anyone prove it.
> >If not, and I conjecture not, what is the largest such number?
> >Anyone with some time (personal and computer) can enjoy this one.
> >-- 
> >
> >Karl Dahlke    ihnp4!ihnet!eklhad
> 
> 
> What *is* that list?  It is not a list of the first n  primes, nor is it a
> list of the primes described...
>                      Muffy

The set is quite definitely finite. In fact it is rather small and takes
very little computer time to generate. A better way of defining it is:

The set of all primes p such that 10p + r  (r = 1,3,7,9) is also in the
set for some r.
 
It is easy to convince yourself that the set should be finite based
on probability arguments.  I don't know of a rigorous proof other
than construction of the set.