22 1. Elementary Prime Number Theory, I
angle determined by e1 and e2, we find that
|(n − 1) + 1/2| = |n − 1/2| ≤
But now (ii) of Theorem 1.10 implies that
N (ξ) =
− n + A
= (n −
+ (n − 1) + A
is prime, so that ξ is a prime element of Z[η] by Lemma 1.13.
A small amount of computation shows that condition (ii) of Theorem
1.10 holds for the values A = 2, 3, 5, 11, 17, and 41. This yields the following
Corollary 1.14. Z[(−1+
D)/2] is a unique factorization domain for D =
−7, −11, −19, −43, −67, −163.
Checking larger values of A does not appear to yield any more examples
satisfying the conditions of Theorem 1.10. Whether or not the list in Corol-
lary 1.14 is complete is known as the class number 1 problem; an equivalent
question appears in Gauss’s Disquisitiones (see [Gau86, Art. 303]). In
1933, Lehmer showed [Leh33] that any missing value of A is necessarily
large, in that |D| 5 ·
In 1934, Heilbronn & Linfoot [HL34] showed
that there is at most one missing value of A. Finally, in 1952, Heegner settled
the problem, using new techniques from the theory of modular functions:
Theorem 1.15 (Heegner). If A 41, then Z[η] does not have unique fac-
torization. Hence if A ≥ 2 is an integer for which
+ n + A is prime for
all 0 ≤ n A − 1, then A ≤ 41.
For a modern account of Heegner’s proof, see [Cox89, §12].
9. Primes represented by general polynomials
The result of the previous section leaves a very natural question unresolved:
Does Euler’s polynomial T
+ T + 41, which does such a marvelous job of
producing primes at the first several natural numbers n, represent infinitely
many primes as n ranges over the set of all positive integers? More generally,
what can one say about the set of prime values assumed by a polynomial
F (T) ∈ Z[T]? In this section we survey the known results in this direction.
9.1. The linear case. Suppose first that F (T) is linear, say F (T) = a +
mT, where m 0. Asking whether F (n) is prime for infinitely many natural
numbers n amounts to asking whether the infinite arithmetic progression
a + m, a + 2m, a + 3m, a + 4m, . . .