🚀 go-pugleaf

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 integers, 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2^31, including an unpublished one furnished by Peter Montgomery,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) .

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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
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