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From: bs@faron.UUCP (Robert D. Silverman)
Newsgroups: net.math
Subject: palindomic primes, one more time
Message-ID: <186@faron.UUCP>
Date: Sat, 8-Dec-84 09:39:59 EST
Article-I.D.: faron.186
Posted: Sat Dec  8 09:39:59 1984
Date-Received: Sun, 16-Dec-84 04:43:17 EST
Organization: The MITRE Corp., Bedford, Ma.
Lines: 24


	Numbers of the form 10**n + 1 can not be prime unless n is
	a power of 2. This is easy to see because if n = pq and (say)
	p is odd then 10**pq + 1 is divisible by 10**q + 1. These
	are the base 10 analog of Fermat numbers 2**(2**n) + 1. There
	are no base 10 Fermat primes other than 101 for n <= 15
	of (10**(2**N)) + 1.

	To PROVE R317 is prime is far from trivial. It is easy to
	show that it is a probable prime. In order to prove it one
	needs to know at least partial factorizations of 10**158 + 1,
	10**79 + 1, and 10**79 - 1, and these latter are difficult
	to obtain. 10**79 + 1 has been completely factored, but
	10**79 -1 still has a composite 63 digit cofactor and only
	a few factors of 10**158 + 1 are known. One also needs to
	invoke methods of Hugh Williams [see previous posting]
	to finish the primaility PROOF.

	By the way, the 9'th Fermat number 2^512+1 is currently one
	of the most wanted factorizations. Anyone want to have a
	crack at it?  It has the small factor 2424833, but the 
	remaining 148 digit cofactor is still composite.