28 1. Elementary Prime Number Theory, I

Conjecture 1.24 (Euler). There are infinitely many natural numbers n for

which

n2

+ 1 is prime.

Similarly, it seems reasonable to conjecture that our old friend, T

2

+T +

41, represents infinitely many primes. Once again, formulating conjectures

of this type requires some care; if

n2

+ 1 or

n2

+ n + 41 is replaced by

n2

+ n + 2, then the statement corresponding to Euler’s conjecture is false,

since

n2

+ n + 2 is always even.

Suppose more generally that F1(T),...,Fr(T) ∈ Z[T] are nonconstant

polynomials, each with positive leading coeﬃcient. We can ask when it is

the case that F1(n),...,Fr(n) are simultaneously prime for infinitely many

natural numbers n. Evidently if this is to be the case, then we must suppose

that each Fi is irreducible over Z. The example of r = 2 and F1(T) = T,

F2(T) = T +1 shows that this is not suﬃcient, as does the example of r = 1

and F1(T) = T

2

+ T + 2. What goes wrong in these examples is that there

is a local obstruction: If we put G(T) :=

r

i=1

Fi(T), then G(n) is always

even. In 1958, Schinzel conjectured (see [SS58]) that these are the only

remaining obstructions to be accounted for:

Conjecture 1.25 (Schinzel’s “Hypothesis H”). Suppose F1(T),...,Fr(T) ∈

Z[T] are nonconstant and irreducible and that each Fi has a positive leading

coeﬃcient. Put G(T) :=

r

i=1

Fi(T), and suppose that there is no prime p

which divides G(n) for every integer n. Then F1(n), F2(n), . . . , Fr(n) are

simultaneously prime for infinitely many natural numbers n.

The hypothesis on G is necessary: Suppose that p is a (fixed) prime

which divides G(n) for each n. Then p divides some Fi(n) for each n. But

for large n, each Fi(n) p, and so for large n, some Fi(n) is composite.

The twin prime conjecture corresponds to choosing r = 2, F1(T) = T,

and F2(T) = T + 2 in Hypothesis H. Taking instead r = 1 and F1(T) =

T

2

+ 1, we recover Euler’s Conjecture 1.24. Despite substantial attention,

both the twin prime conjecture and Conjecture 1.24 remain open. Even

more depressing, no case of Hypothesis H has ever been shown to hold except

when r = 1 and F1(T) is linear, when Hypothesis H reduces to Dirichlet’s

theorem!

Sieve methods, which we introduce in Chapter 6, can be used to obtain

certain approximations to Hypothesis H. We give two examples: A theorem

of Chen [Che73] asserts that there are infinitely many primes p for which

p + 2 is either prime or the product of two primes. And Iwaniec [Iwa78]

has shown that there are infinitely many n for which

n2

+ 1 is either prime

or the product of two primes. (This latter result applies also to

n2

+ n + 41,

and in fact to any quadratic obeying the conditions of Hypothesis H.)