David wrote:

> I incline to the Crandall-Pomerance heuristic

> that there is an infinitude, with O(log(log(N)))

> less than N. (CP Section 1.3.3.)

Me too. I have also been planning to ask Chris. In fact I have a whole file

with comments to the Prime Pages. I should probably get around to actually

mailing it to him one day before it all becomes obsolete. Below is what it has

said for a long time.

http://primes.utm.edu/glossary/xpage/WieferichPrime.html says:

"Are there infinitely many Wieferich primes? Probably not"

I wonder what "Probably not" is based on. The Crandall, Dilcher and Pomerance

reference says on page 14 of

http://www.math.dartmouth.edu/~carlp/PDF/paper111.pdf:
"Thus, heuristically we might argue that the number of Wieferich primes in an

interval [x; y] is expected to be sum(1/p) over x<=p<=y ~= ln(ln y/ln x)."

This is the same as the estimate on

http://primes.utm.edu/glossary/xpage/WilsonPrime.html
(The reason for the estimates is the same: A guess that each p has chance 1/p)

--

Jens Kruse Andersen