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From: bill@ur-cvsvax.UUCP (Bill Vaughn)
Newsgroups: net.math
Subject: Re: palindromic prime numbers -- a curious query
Message-ID: <124@ur-cvsvax.UUCP>
Date: Fri, 16-Nov-84 01:00:09 EST
Article-I.D.: ur-cvsva.124
Posted: Fri Nov 16 01:00:09 1984
Date-Received: Sat, 17-Nov-84 07:57:47 EST
References: <3470@ecsvax.UUCP>
Organization: Center for Visual Science, U. of Rochester
Lines: 41

11 is the only palindromic prime with an even number of digits. All other
even palindromic numbers are divisible by 11. This is because the
alternating sum of the digits of a palindromic number with an even number
of digits is 0, and this is a well knwon test for divisibilty by 11.

As for Dr. Ziff's conjecture, I think that it is highly unlikely.  I
probably have a counter-example of some 9 digit palindromic prime somewhere
in some research I did some years ago, but I can't put my hands on it
right at this moment. I did find these facts however: the number of n-digit
palindromic numbers is 9*10^[(n-1)/2]. (PP = palindromic primes).
There are 15 3 digit PP's out of 90 possibilities. ~17%
  "    "  93 5 digit PP's  "  "  900     "         ~10%
  "    " 668 7 digit PP's  "  "  9000    "         ~7%

Even though I don't have the number of 9 digit PP's out of the 90,000
possibilities, it just doesn't seem possible that they could buck this
trend and go to zero percent.

Some of the invetigations I carried out years ago showed that even if
one restricted oneself to palindromic's which use only 2 different digits
(i.e 18181 35353 7474747474 etc.) there are still plenty of primes out
there.  I only used Fermat's test with several different bases however,
so the best I can say is that there are plenty of pseudo-primes for as
far as you care to look.  I think I went up to about 19-21 digits.
These ramblings lead me to ask a question which has been nagging me for
some time:

Are there an infinite number of palindromic primes?  If so, can someone
point me to a proof or sketch one out.

Would this proof(if it exists) apply to subsets of PP's such as those which
use only 2 different digits?

What about other bases?

Bill Vaughn
Univ. of Rochester
Center for Visual Science


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