Newsgroups: bit.listserv.nmbrthry Date: Tue, 10 Nov 1992 13:41:57 EST Reply-To: "William F. Hammond" Sender: Number Theory List From: "William F. Hammond" Subject: Primes like 40487 Comments: To: NmbrThry@vm1.nodak.edu To: Multiple recipients of list NMBRTHRY My unverified computations with PARI [an excellent computer library slanted toward number theory, by C. Batut, D. Bernardi, H. Cohen and M. Olivier] show that 40487 is the only odd prime p smaller than 2^31 (about 2.147*10^9) for which the smallest positive primitive root c is NOT a topological generator of the p-adic units, i.e., for which c is a solution of the congruence c^(p-1) = 1 mod p^2. Does anyone else know other primes having this property? I have looked at several lists of solutions of the congruence a^(p-1) = 1 mod p^2 for small values of a with p quite a bit larger than 2^31, including an unpublished one furnished by Peter Montgomery, without finding another example. If the phenomenon known as "approximation" makes it reasonable to view the selection of c as a random selection among the various primitive roots mod p^2 for p in a finite set of primes, then it is reasonable to conjecture that there should be infinitely many of these primes. The same line of reasoning would suggest that the probability of one's not finding such a prime between x and y (with x < y) is about log(x)/log(y) . ---------------------------------------------------------------------- William F. Hammond Dept. of Mathematics & Statistics 518-442-4625 SUNYA, Albany, NY 12222 (U.S.A.) hammond@math.albany.edu FAX: 518-442-4731 ----------------------------------------------------------------------