28 1. Elementary Prime Number Theory, I
Conjecture 1.24 (Euler). There are infinitely many natural numbers n for
+ 1 is prime.
Similarly, it seems reasonable to conjecture that our old friend, T
+T +
41, represents infinitely many primes. Once again, formulating conjectures
of this type requires some care; if
+ 1 or
+ n + 41 is replaced by
+ n + 2, then the statement corresponding to Euler’s conjecture is false,
+ 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
+ T + 2. What goes wrong in these examples is that there
is a local obstruction: If we put G(T) :=
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) :=
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) =
+ 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
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
+ 1 is either prime
or the product of two primes. (This latter result applies also to
+ n + 41,
and in fact to any quadratic obeying the conditions of Hypothesis H.)
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