Schinzel's hypothesis H

In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the maximum possible scope of a conjecture of the nature that a family

:"f_i"("n")

of values of irreducible polynomials "f"("t") should be able to take on prime number values simultaneously, for an integer "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

:"f_i"("n")

are prime numbers.

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

This can be done by means of the concept of integer-valued polynomial. This allows us to say that 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" + 4)("x" + 7)

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

:"Q"("x") = Π "f_i"("x")

for any conjecture, since their presence is quickly seen to contradict the possibility that "f_i"("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 "f_i"("n") will be simultaneously prime, infinitely often, for any choice of irreducible integral polynomials "f_i"("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.

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 H_N) 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 "f_i"("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) thatthe polynomial

::$x^8\; +\; u^3,$

over the ring $F\_2\; [u]\; ,$ is irreducible and has no fixed prime polynomial divisor (after all, its values at "x" = 0 and "x" = 1 are relatively prime polynomials) but allof its values as "x" runs over $F\_2\; [u]\; ,$ are composite. Similar examples canbe found with $F\_2,$ replaced by any finite field; the obstructions in a properformulation of Hypothesis H over "F" ["u"] , where "F" is a finite field, are nolonger just "local" but a new "global" obstruction occurs with no classical parallel.

External links

* [http://www.impan.gov.pl/User/schinzel/] for the publications of the Polish mathematician Andrzej Schinzel. The hypothesis derives from paper 25 on that list, from 1958, written with Sierpiński.