Prime Numbers

1.3 Primes of special form 31
argument is made as with Mersenne primes, we get that the number of
Fermat primes is finite. This comes from the convergence of the sum of n/2 n ,
which expression one finds is proportional to the supposed probability that
Fn is prime. If this kind of heuristic is to be taken seriously, it suggests that
there are no more Fermat primes after F4, the point where Fermat stopped,
confidently predicting that all larger Fermat numbers are prime! A heuristic
suggested by H. Lenstra, similar in spirit to the previous estimate on the
density of Mersenne primes, says that the “probability” that Fn is prime is
approximately
e γ lg b
, (1.13)
2n where b is the current limit on the possible prime factors of Fn. If nothing is
known about possible factors, one might use the smallest possible lower bound
b =3·2n+2 +1 for the numerator calculation, giving a rough a priori probability
of n/2n that Fn is prime. (Incidentally, a similar probability argument for
generalized Fermat numbers b2n + 1 appears in [Dubner and Gallot 2002].) It
is from such a probabilistic perspective that Fermat’s guess looms as ill-fated
as can be.
1.3.3 Certain presumably rare primes
There are interesting classes of presumably rare primes. We say “presumably”
because little is known in the way of rigorous density bounds, yet empirical
evidence and heuristic arguments suggest relative rarity. For any odd prime p,
Fermat’s “little theorem” tells us that 2 p−1 ≡ 1(modp). One might wonder
whether there are primes such that
2 p−1 ≡ 1(modp 2 ), (1.14)
such primes being called Wieferich primes. These special primes figure strongly
in the so-called first case of Fermat’s “last theorem,” as follows. In [Wieferich
1909] it is proved that if
x p + y p = z p ,
where p is a prime that does not divide xyz, thenp satisfies relation (1.14).
Equivalently, we say that p is a Wieferich prime if the Fermat quotient
qp(2) = 2p−1 − 1
p
vanishes (mod p). One might guess that the “probability” that qp(2) so
vanishes is about 1/p. Since the sum of the reciprocals of the primes is
divergent (see Exercise 1.20), one might guess that there are infinitely many
Wieferich primes. Since the prime reciprocal sum diverges very slowly, one
might also guess that they are very few and far between.
The Wieferich primes 1093 and 3511 have long been known. Crandall,
Dilcher, and Pomerance, with the computational aid of Bailey, established
that there are no other Wieferich primes below 4 · 10 12 [Crandall et al. 1997].