Prime Numbers

1.2 Celebrated conjectures and curiosities 15
events,” then the probability they are both prime should be about 1/ ln 2 x.
Thus, if we denote the twin-prime-pair counting function by
π2(x) =#{p ≤ x : p, p +2∈P},
where P is the set of all primes, then we might guess that
π2(x) ∼
x
2
1
ln 2 t dt.
However, it is somewhat dishonest to consider p and p + 2 as independent
prime events. In fact, the chance of one being prime influences the chance
that the other is prime. For example, since all primes p>2 are odd, the
number p + 2 is also odd, and so has a “leg up” on being prime. Random
odd numbers have twice the chance of being prime as a random number not
stipulated beforehand as odd. But being odd is only the first in a series of
“tests” a purported prime must pass. For a fixed prime q, a large prime must
pass the “q-test” meaning “not divisible by q.” If p is a random prime and
q>2, then the probability that p+2 passes the q-test is (q−2)/(q−1). Indeed,
from (1.5), there are ϕ(q) =q − 1 equally likely residue classes modulo q for p
to fall in, and for exactly q−2 of these residue classes we have p+2 not divisible
by q. But the probability that a completely random number passes the q-test
is (q − 1)/q. So, let us revise the above heuristic with the “fudge factor” 2C2,
where C2 =0.6601618158 ... is the so-called “twin-prime constant”:
C2 =
2