Prime Numbers

1.2 Celebrated conjectures and curiosities 17
to the implied precision. The estimate was made by computing the reciprocal
sum very accurately for twin primes up to 10 16 and then extrapolating to the
infinite sum using (1.7) to estimate the tail of the sum. (All that is actually
proved rigorously about B ′ (by year 2004) is that it is between a number
slightly larger than 1.83 and a number slightly smaller than 2.347.) In his
earlier (1995) computations concerning the Brun constant, Nicely discovered
the now-famous floating-point flaw in the Pentium computer chip, a discovery
that cost the Pentium manufacturer Intel millions of dollars. It seems safe to
assume that Brun had no idea in 1909 that his remarkable theorem would
have such a technological consequence!
1.2.2 Prime k-tuples and hypothesis H
The twin prime conjecture is actually a special case of the “prime k-tuples”
conjecture, which in turn is a special case of “hypothesis H.” What are these
mysterious-sounding conjectures?
The prime k-tuples conjecture begins with the question, what conditions
on integers a1,b1,...,ak,bk ensure that the k linear expressions a1n +
b1,...,akn + bk are simultaneously prime for infinitely many positive integers
n? One can see embedded in this question the first part of the Dirichlet
Theorem 1.1.5, which is the case k = 1. And we can also see embedded the
twin prime conjecture, which is the case of two linear expressions n, n +2.
Let us begin to try to answer the question by giving necessary conditions
on the numbers ai,bi. We rule out the cases when some ai = 0, since such a
case collapses to a smaller problem. Then, clearly, we must have each ai > 0
and each gcd(ai,bi) = 1. This is not enough, though, as the case n, n +1
quickly reveals: There are surely not infinitely many integers n for which n
and n + 1 are both prime! What is going on here is that the prime 2 destroys
the chances for n and n+1,sinceoneofthemisalwayseven,andevennumbers
are not often prime. Generalizing, we see that another necessary condition is
that for each prime p there is some value of n such that none of ain + bi
is divisible by p. This condition automatically holds for all primes p>k;
it follows from the condition that each gcd(ai,bi) =1.Theprimek-tuples
conjecture [Dickson 1904] asserts that these conditions are sufficient:
Conjecture 1.2.1 (Prime k-tuples conjecture). If a1,b1,...,ak,bk are integers
with each ai > 0, each gcd(ai,bi) =1, and for each prime p ≤ k, there
is some integer n with no ain+bi divisible by p, then there are infinitely many
positive integers n with each ain + bi prime.
Whereas the prime k-tuples conjecture deals with linear polynomials,
Schinzel’s hypothesis H [Schinzel and Sierpiński 1958] deals with arbitrary
irreducible polynomials with integer coefficients. It is a generalization of
an older conjecture of Bouniakowski, who dealt with a single irreducible
polynomial.