In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory.
The hypothesis claims that for every finite collection
of nonconstant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds: The second condition is satisfied by sets such as
No effective technique is known for determining whether the first condition holds for a given set of polynomials, but the second one is straightforward to check: Let
and compute the greatest common divisor of
One can see by extrapolating with finite differences that this divisor will also divide all other values of
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
The hypothesis then implies the existence of infinitely many twin primes, a basic and notorious open problem.
As proved by Schinzel and Sierpiński[1] it is equivalent to the following: if condition 2 does not hold, then there exists at least one positive integer
will be simultaneously prime, for any choice of irreducible integral polynomials
The special case of a single linear polynomial is Dirichlet's theorem on arithmetic progressions, one of the most important results of number theory.
In fact, this special case is the only known instance of Schinzel's Hypothesis H. We do not know the hypothesis to hold for any given polynomial of degree greater than
Almost prime approximations to Schinzel's Hypothesis have been attempted by many mathematicians; among them, most notably, Chen's theorem states that there exist infinitely many prime numbers
is either a prime or a semiprime [2] and Iwaniec proved that there exist infinitely many integers
[3] Skorobogatov and Sofos have proved that almost all polynomials of any fixed degree satisfy Schinzel's hypothesis H.[4] Let
be an integer-valued polynomial with common factor
Ronald Joseph Miech proved using Brun sieve that
The proof of the Miech's theorem uses Brun sieve.
If there is a hypothetical probabilistic density sieve, using the Miech's theorem can prove the Schinzel's hypothesis H in all cases by mathematical induction.
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.
This connection is due to Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc.
[5] For further explanations and references on this connection see the notes of Swinnerton-Dyer.
[6] The conjectural result being so strong in nature, it is possible that it could be shown to be too much to expect.
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 that has no fixed divisor > 1.
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.
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.