Schinzel's hypothesis H
Encyclopedia
In mathematics
, Schinzel's hypothesis H is a very broad generalisation of conjecture
s 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
with values at integers n of irreducible
integer-valued polynomial
s
should be able to take on prime number
values simultaneously, for integers n that can be as large as we please. Putting it another way, there should be infinitely many such n, for which each of the sequence values are prime numbers. Some constraints are needed on the polynomials. Andrzej Schinzel
's hypothesis builds on the earlier Bunyakovsky conjecture
, for a single polynomial.
is also an integer-valued polynomial. For example, we can say that
(x + 7)
has 2 as fixed divisor. Such fixed divisors must be ruled out of
for any conjecture for polynomials fi, i = 1 to k, since their presence is quickly seen to contradict the possibility that fi(n) can all be prime, with large values of n.
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 n2+1 is often prime for n up to 1500.
, 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.
, but a closely related version (hypothesis HN) does. That requires an extra polynomial F(x), which in the Goldbach problem would just be x, for which
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
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).
with no local obstruction to taking infinitely many prime values is conjectured to take infinitely many prime values.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Schinzel's hypothesis H is a very broad generalisation of conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
s 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
- f(n), g(n), ...
with values at integers n of irreducible
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
integer-valued polynomial
Integer-valued polynomial
In mathematics, an integer-valued polynomial P is a polynomial taking an integer value P for every integer n. Certainly every polynomial with integer coefficients is integer-valued. There are simple examples to show that the converse is not true: for example the polynomialgiving the triangle...
s
- f(t), g(t), ...
should be able to take on prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
values simultaneously, for integers n that can be as large as we please. Putting it another way, there should be infinitely many such n, for which each of the sequence values are prime numbers. Some constraints are needed on the polynomials. Andrzej Schinzel
Andrzej Schinzel
Andrzej Bobola Maria Schinzel is a Polish mathematician, studying mainly number theory.- Biography :Schinzel received his Ph.D...
's hypothesis builds on the earlier Bunyakovsky conjecture
Bunyakovsky conjecture
The Bunyakovsky conjecture stated in 1857 by the Russian mathematician Viktor Bunyakovsky, claims that an irreducible polynomial of degree two or higher with integer coefficients generates for natural arguments either an infinite set of numbers with greatest common divisor exceeding unity, or...
, for a single polynomial.
Necessary limitations
Such a conjecture must be subject to some necessary conditions. For example if we take the two polynomials x + 4 and x + 7, there is no n > 0 for which n + 4 and n + 7 are both primes. That is because one will be an even number > 2, and the other an odd number. The main question in formulating the conjecture is to rule out this phenomenon.Fixed divisors pinned down
The arithmetic nature of the most evident necessary conditions can be understood. An integer-valued polynomial Q(x) has a fixed divisor m if there is an integer m > 1 such that- Q(x)/m
is also an integer-valued polynomial. For example, we can say that
(x + 7)
has 2 as fixed divisor. Such fixed divisors must be ruled out of
- Q(x) = Π fi(x)
for any conjecture for polynomials fi, i = 1 to k, since their presence is quickly seen to contradict the possibility that fi(n) can all be prime, with large values of n.
Formulation of hypothesis H
Therefore the standard form of hypothesis H is that if Q defined as above has no fixed prime divisor, then all fi(n) will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials fi(x) with positive leading coefficients.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,
- x2 + 1
has no fixed prime divisor. We therefore expect that there are infinitely many primes
- n2 + 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 n2+1 is often prime for n up to 1500.
Prospects and applications
The hypothesis is probably not accessible with current methods in analytic number theoryAnalytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
, but is now quite often used to prove conditional results, for example in diophantine geometry
Diophantine geometry
In mathematics, diophantine geometry is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry over a ground field K that is not algebraically closed, such as the field of rational numbers or a finite field, or more general...
. 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 conjectureGoldbach's conjecture
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states:A Goldbach number is a number that can be expressed as the sum of two odd primes...
, but a closely related version (hypothesis HN) does. That requires an extra polynomial F(x), which in the Goldbach problem would just be 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 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 polynomialswith 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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, are no longer just local but a new global obstruction occurs with no classical parallel, assuming hypothesis H is in fact correct.
External links
- http://www.impan.gov.pl/User/schinzel/ for the publications of the Polish mathematician Andrzej SchinzelAndrzej SchinzelAndrzej Bobola Maria Schinzel is a Polish mathematician, studying mainly number theory.- Biography :Schinzel received his Ph.D...
. The hypothesis derives from paper 25 on that list, from 1958, written with Sierpiński.