Prime Numbers

1.2 Celebrated conjectures and curiosities 21
It sounds straightforward enough, and perhaps it is, but it also may be that
any value of x required to demolish the convexity conjecture is enormous. (See
Exercise 1.92 for more on such issues.)
1.2.5 Prime-producing formulae
Prime-producing formulae have been a popular recreation, ever since the
observation of Euler that the polynomial
x 2 + x +41
attains prime values for each integer x from0to39inclusive.Armedwith
modern machinery, one can empirically analyze other polynomials that give,
over certain ranges, primes with high probability (see Exercise 1.17). Here
are some other curiosities, of the type that have dubious value for the
computationalist (nevertheless, see Exercises 1.5, 1.77):
Theorem 1.2.2 (Examples of prime-producing formulae). There exists a
real number θ>1 such that for every positive integer n, the number
θ 3n
is prime. There also exists a real number α such that the n-th prime is given
by:
pn = 10 2n+1
2n
α − 10 10 2n
α .
This first result depends on a nontrivial theorem on the distribution of primes
in “short” intervals [Mills 1947], while the second result is just a realization of
the fact that there exists a well-defined decimal expansion α = pm10−2m+1. Such formulae, even when trivial or almost trivial, can be picturesque.
By appeal to the Wilson theorem and its converse (Theorem 1.3.6), one may
show that
n
(j − 1)!+1 (j − 1)!
π(n) =
−
,
j
j
j=2
but this has no evident value in the theory of the prime-counting function
π(n). Yet more prime-producing and prime-counting formulae are exhibited
in the exercises.
Prime-producing formulae are often amusing but, relatively speaking,
useless. There is a famous counterexample though. In connection with the
ultimate resolution of Hilbert’s tenth problem, which problem asks for a
deterministic algorithm that can decide whether a polynomial in several
variables with integer coefficients has an all integral root, an attractive
side result was the construction of a polynomial in several variables with
integral coefficients, such that the set of its positive values at positive integral
arguments is exactly the set of primes (see Section 8.4).