bs@faron.UUCP (Robert D. Silverman) (12/08/84)
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.