In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the possible scope of a conjecture of the nature that several sequences of the type
Contents
- Necessary limitations
- Fixed divisors pinned down
- Formulation of hypothesis H
- Prospects and applications
- Extension to include the Goldbach conjecture
- Local analysis
- An analogue that fails
- References
with values at integers
should be able to take on prime number values simultaneously, for integers
Necessary limitations
Such a conjecture must be subject to some necessary conditions. For example, if we take the two polynomials
Thus, we should add a condition: "For every prime p, there is a n such that all the polynomial values at n are not divisible by p".
Fixed divisors pinned down
The arithmetic nature of the most evident necessary conditions can be understood. An integer-valued polynomial
is also an integer-valued polynomial. For example, we can say that
has 2 as fixed divisor. Such fixed divisors must be ruled out of
for any conjecture for polynomials
Formulation of hypothesis H
Therefore, the standard form of hypothesis H is that if
If the leading coefficients were negative, we could expect negative prime values; this is a harmless restriction, really. There is probably no real reason to restrict to integral polynomials, rather than integer-valued polynomials. The condition of having no fixed prime divisor is certainly effectively checkable in a given case, since there is an explicit basis for the integer-valued polynomials. As a simple example,
has no fixed prime divisor. We therefore expect that there are infinitely many primes
This has not been proved, though. It was one of Landau's conjectures and goes back to Euler, who observed in a letter to Goldbach in 1752 that
Prospects and applications
The hypothesis is probably not accessible with current methods in analytic number theory, but is now quite often used to prove conditional results, for example in diophantine geometry. The conjectural result being so strong in nature, it is possible that it could be shown to be too much to expect.
Extension to include the Goldbach conjecture
The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis HN) does. That requires an extra polynomial
is required to be a prime number, also. This is cited in Halberstam and Richert, Sieve Methods. The conjecture here takes the form of a statement when N is sufficiently large, and subject to the condition
Q(n)(N − F(n))has no fixed divisor > 1. Then we should be able to require the existence of n such that N − F(n) is both positive and a prime number; and with all the fi(n) prime numbers.
Not many cases of these conjectures are known; but there is a detailed quantitative theory (Bateman–Horn conjecture).
Local analysis
The condition of having no fixed prime divisor is purely local (depending just on primes, that is). In other words, a finite set of irreducible integer-valued polynomials with no local obstruction to taking infinitely many prime values is conjectured to take infinitely many prime values.
An analogue that fails
The analogous conjecture with the integers replaced by the one-variable polynomial ring over a finite field is false. For example, Swan noted in 1962 (for reasons unrelated to Hypothesis H) that the polynomial
over the ring F2[u] is irreducible and has no fixed prime polynomial divisor (after all, its values at x = 0 and x = 1 are relatively prime polynomials) but all of its values as x runs over F2[u] are composite. Similar examples can be found with F2 replaced by any finite field; the obstructions in a proper formulation of Hypothesis H over F[u], where F is a finite field, are no longer just local but a new global obstruction occurs with no classical parallel, assuming hypothesis H is in fact correct.