7. Prime-producing formulas 13
Proof. We use the theorem that a Dedekind domain with finitely many
nonzero prime ideals is a principal ideal domain (see, e.g., [Lor96, Propo-
sition III.2.12]) and thus also a unique factorization domain. The ring of
integers OK of a number field K is always a Dedekind domain; consequently,
if K does not possess unique factorization, then OK has infinitely many
nonzero prime ideals. Each such prime ideal lies above a rational prime p,
and for each prime p there are at most [K : Q] prime ideals lying above it.
It follows that there are infinitely many primes p, provided that there is a
single number field K for OK does not possess unique factorization.
And there is: If K = Q(
−5), then
6 = (1 +


is a well-known instance of the failure of unique factorization in OK =

7. Prime-producing formulas
A mathematician is a conjurer who gives away his se-
crets. J. H. Conway
Now that we know there are infinitely many primes, the next question is:
Where are they hiding? Or, to ask a question that has ensnared many
who have flirted with number theory: Is there a formula for producing
primes? This line of inquiry, as natural as it seems, has not been very
The following 1952 result of Sierpi´ nski [Sie52] is representative of many
in this subject. Let pn denote the nth prime number. Define a real number
ξ by putting
ξ :=

= 0.02030005000000070000000000000011 . . ..
Theorem 1.7. We have
pn =
102n−1 102n−1
ξ .
This is, in the literal sense, a formula for primes. But while it may
have some aesthetic merit, it must be considered a complete failure from
the standpoint of utility; determining the number ξ seems to require us
to already know the sequence of primes. A similar criticism can be leveled
against a result of Mills [Mil47], which asserts the existence of a real number
A 1 with the property that
is prime for each natural number n.
A more surprising way of generating primes was proposed by J. H. Con-
way [Con87]. Consider the following list of 14 fractions:
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