Prime Numbers

26 Chapter 1 PRIMES!
Mersenne conjecture, that any two of the following implies the third: (a) the
prime q is either 1 away from a power of 2, or 3 away from a power of 4, (b)
2q − 1isprime,(c)(2q +1)/3 is prime. Once one gets beyond small numbers,
it is very difficult to find any primes q that satisfy two of the statements, and
probably there are none beyond 127. That is, probably the conjecture is true,
but so far it is only an assertion based on a very small set of primes.
It has also been conjectured that every Mersenne number Mq, withq
prime, is squarefree (which means not divisible by a square greater than 1),
but we cannot even show that this holds infinitely often. Let M denote the
set of primes that divide some Mq with q prime. We know that the number
of members of M up to x is o(π(x)), and it is known on the assumption of
the generalized Riemann hypothesis that the sum of the reciprocals of the
members of M converges [Pomerance 1986].
It is possible to give a heuristic argument that supports the assertion that
there are ∼ c ln x primes q ≤ x with Mq prime, where c = eγ / ln 2 and γ is
Euler’s constant. For example, this formula suggests that there should be, on
average, about 23.7 values of q in an interval [x, 10000x]. Assuming that the
machine checks of the Mersenne exponents up to 12000000 are exhaustive,
the actual number of values of q with Mq prime in [x, 10000x] is 23, 24, or
25 for x = 100, 200,...,1200, with the count usually being 24. Despite the
good agreement with practice, some still think that the “correct” value of c
is 2/ ln 2 or something else. Until a theorem is actually proved, we shall not
know for sure.
We begin the heuristic with the fact that the probability that a random
number near Mq =2q− 1 is prime is about 1/ ln Mq, as seen by the prime
number Theorem 1.1.4. However, we should also compare the chance of Mq
being prime with a random number of the same size. It is likely not the same,
as Theorem 1.3.2 already indicates. Let us ignore for a moment the intricacies
of this theorem and use only that Mq has no prime factors in the interval
[1,q]. Here q is about lg Mq (here and throughout the book, lg means log2). What is the chance that a random number near x whose least prime factor
exceeds lg x is prime? We know how to answer this question rigorously. First
consider the chance that a random number near x has its least prime factor
exceeding lg x. Intuitively, this probability should be
P :=
1 − 1
,
p
p≤lg x
since each prime p has probability 1/p of dividing a random number, and
these should be at least roughly independent events. They cannot be totally
independent, for example, no number in [1,x] is divisible by two primes in the
interval (x 1/2 ,x], yet a purely probabilistic argument suggests that a positive
proportion of the numbers in [1,x] actually have this property! However, when
dealing with very small primes, and in this case only those up to lg x, the
heuristic guess is provable. Now, each prime near x survives this sieve; that is,
it is not divisible by any prime p ≤ lg x. So, if a number n near x has already