Prime Numbers

1.3 Primes of special form 27
passed this lg x sieve, then its probability of being prime should climb from
1/ ln x to
1
P ln x .
We know P asymptotically. It follows from the Mertens theorem (see Theorem
1.4.2) that 1/P ∼ e γ ln lg x as x →∞. Thus, one might conclude that Mq is
prime with “probability” e γ ln lg Mq/ ln Mq. But this expression is very close to
e γ ln q/(q ln 2). Summing this expression for primes q ≤ x, we get the heuristic
asymptotic expression for the number of Mersenne prime exponents up to x,
namely c ln x with c = e γ / ln 2.
If one goes back and tries to argue in a more refined way using Theorem
1.3.2, then one needs to use not only the fact that the possible prime factors of
Mq are quite restricted, but also that a prime that meets the condition of this
theorem has an enhanced chance of dividing Mq. For example, if p = kq +1 is
prime and p ≡±1 (mod 8), then one might argue that the chance that p|Mq
is not 1/p, but rather the much larger 2/k. It seems that these two criteria
balance out, that is, the restricted set of possible prime factors balances with
the enhanced chance of divisibility by them, and we arrive at the same estimate
as above. This more difficult argument was presented in the first edition of
this book.
1.3.2 Fermat numbers
The celebrated Fermat numbers Fn =22n+1, like the Mersenne numbers, have
been the subject of much scrutiny for centuries. In 1637 Fermat claimed that
the numbers Fn arealwaysprime,andindeedthefirstfive,uptoF4 = 65537
inclusive, are prime. However, this is one of the few cases where Fermat was
wrong, perhaps very wrong. Every other single Fn for which we have been
able to decide the question is composite! The first of these composites, F5,
was factored by Euler.
A very remarkable theorem on prime Fermat numbers was proved by
Gauss, again from his teen years. He showed that a regular polygon with n
sides is constructible with straightedge and compass if and only if the largest
odd divisor of n isaproductofdistinctFermatprimes.IfF0,...,F4 turn out
to be the only Fermat primes, then the only n-gons that are constructible are
those with n =2am with m|232 − 1 (since the product of these five Fermat
primes is 232 − 1).
If one is looking for primes that are 1 more than a power of 2, then one
need look no further than the Fermat numbers:
Theorem 1.3.4. If p =2 m +1 is an odd prime, then m is a power of two.
Proof. Assume that m = ab, wherea is the largest odd divisor of m. Thenp
has the factor 2 b + 1. Therefore, a necessary condition that p beprimeisthat
p =2 b +1;thatis,a = 1 and m = b is a power of 2. ✷