Schinzel's hypothesis H

The hypothesis aims to define the possible scope of a conjecture of the nature that several sequences of the type

${\displaystyle f(n),g(n),\ldots ,}$

with values at integers ${\displaystyle n}$ of irreducible integer-valued polynomials

${\displaystyle f(x),g(x),\ldots ,}$

should be able to take on prime number values simultaneously, for arbitrarily large integers ${\displaystyle n}$. Putting it another way, there should be infinitely many such ${\displaystyle n}$ for which each of the sequence values are prime numbers. Some constraints are needed on the polynomials. Schinzel's hypothesis builds on the earlier Bunyakovsky conjecture, for a single polynomial, and on the Hardy–Littlewood conjectures and Dickson's conjecture for multiple linear polynomials. It is in turn extended by the Bateman–Horn conjecture.

Note that the coefficients of the polynomials need not to be integers; for example, this conjecture includes the polynomial ${\displaystyle {\frac {1}{2}}x^{2}+{\frac {1}{2}}x+1}$, since it is an integer-valued polynomial.

Such a conjecture requires necessary conditions. For example, if we take the two polynomials ${\displaystyle x+4}$ and ${\displaystyle x+7}$, there is no ${\displaystyle n>0}$ for which ${\displaystyle n+4}$ and ${\displaystyle n+7}$ are both primes. That is because one of them will be an even number ${\displaystyle n>2}$. The main question in formulating the conjecture is to rule out this phenomenon.

Thus, we should add a condition: "For every prime p, there is an n such that all the polynomial values at n are not divisible by p".

The arithmetic nature of the most evident necessary conditions can be understood. An integer-valued polynomial ${\displaystyle Q(x)}$ has a fixed divisor ${\displaystyle m}$ if there is an integer ${\displaystyle m>1}$ such that

${\displaystyle {\frac {Q(x)}{m}}}$

is also an integer-valued polynomial. For example, we can say that

${\displaystyle (x+4)(x+7)}$

has 2 as fixed divisor. Such fixed divisors must be ruled out of

${\displaystyle Q(x)=\prod _{i=1}^{k}f_{i}(x)}$

for any conjecture for polynomials ${\displaystyle f_{i}}$, ${\displaystyle i=1,\ldots ,k}$, since their presence is quickly seen to contradict the possibility that ${\displaystyle f_{i}(n)}$ can all be prime, with large values of ${\displaystyle n}$.

Therefore, the standard form of Schinzel's hypothesis H is that if ${\displaystyle Q}$ defined as above has no fixed prime divisor, then all ${\displaystyle f_{i}(n)}$ will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials ${\displaystyle f_{i}(x)}$ with positive leading coefficients.

As proved by Schinzel and Sierpiński in page 188 of [1] it is equivalent to the following: if ${\displaystyle Q}$ defined as above has no fixed prime divisor, then there exists at least one positive integer ${\displaystyle n}$ such that all ${\displaystyle f_{i}(n)}$ will be simultaneously prime, for any choice of irreducible integral polynomials ${\displaystyle f_{i}(x)}$ with positive leading coefficients.

If the leading coefficients were negative, we could expect negative prime values; this is a harmless restriction. 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,

${\displaystyle x^{2}+1}$

has no fixed prime divisor. We therefore expect that there are infinitely many primes

${\displaystyle n^{2}+1}$

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 ${\displaystyle n^{2}+1}$ is often prime for ${\displaystyle n}$ up to 1500.

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 ${\displaystyle 1}$, nor for any system of more than one polynomial. To get a sense of how hard the hypothesis is, note that in the very small special case ${\displaystyle f_{1}(t)=t,f_{2}(t)=t+2}$, the hypothesis implies the existence of infinitely many twin primes, a basic and notorious open problem.

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 numbers ${\displaystyle n}$ such that ${\displaystyle n+2}$ is either a prime or a semiprime [2] and Iwaniec proved that there exist infinitely many integers ${\displaystyle n}$ for which ${\displaystyle n^{2}+1}$ is either a prime or a semiprime.[3] Skorobogatov and Sofos have proved that almost all polynomials of any fixed degree satisfy Schinzel's hypothesis H.[4]

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 [6] of Swinnerton-Dyer. The conjectural result being so strong in nature, it is possible that it could be shown to be too much to expect.

The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis H_N) does. That requires an extra polynomial ${\displaystyle F(x)}$, which in the Goldbach problem would just be ${\displaystyle x}$, for which

N − F(n)

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 f_i(n) prime numbers.

Not many cases of these conjectures are known; but there is a detailed quantitative theory (Bateman–Horn conjecture).

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.

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

${\displaystyle x^{8}+u^{3}\,}$

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.

• Crandall, Richard; Pomerance, Carl B. (2005). Prime Numbers: A Computational Perspective (Second ed.). New York: Springer-Verlag. doi:10.1007/0-387-28979-8. ISBN 0-387-25282-7. MR 2156291. Zbl 1088.11001.
• Guy, Richard K. (2004). Unsolved problems in number theory (Third ed.). Springer-Verlag. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Pollack, Paul (2008). "An explicit approach to hypothesis H for polynomials over a finite field". In De Koninck, Jean-Marie; Granville, Andrew; Luca, Florian (eds.). Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. CRM Proceedings and Lecture Notes. 46. Providence, RI: American Mathematical Society. pp. 259–273. ISBN 978-0-8218-4406-9. Zbl 1187.11046.
• Swan, R. G. (1962). "Factorization of Polynomials over Finite Fields". Pacific Journal of Mathematics. 12 (3): 1099–1106. doi:10.2140/pjm.1962.12.1099.

• for the publications of the Polish mathematician Andrzej Schinzel. The hypothesis derives from paper,[1] which is paper 25 on that list, from 1958, written with Sierpiński.