Schinzels hypothesis H

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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

Pi(n)

of values of irreducible polynomials P(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

Pi(n)

are prime numbers.

Such a conjecture must be subject to some necessary conditions. For example if we take the two polynomials t+4 and t+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.

This can be done by means of the concept of integer-valued polynomial. This allows us to say that an integer-valued polynomial Q(t) has a fixed divisor m if there is an integer m > 0 such that

Q(t)/m

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

(t+4)(t+7)

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

Q(t) = Π Pi(t)

for any conjecture, since their presence is quickly seen to contradict the possibility that Pi(n) can all be prime, with large values of n.

Therefore the standard form of hypothesis H is that if Q defined as above has no fixed prime divisor, then the Pi(n) will be simultaneously prime, infinitely often, for any choice of integral polynomials Pi(t) with positive leading coefficients.

If the leading coefficients were negative, we could at most expect values −p with p prime; this is a harmless restriction, really. There is probably no real reason to restrict to integral polynomials, rather than integer 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,

t2 + 1

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

n2 + 1.

This has not been proved, though.

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.

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

NF(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 NF(n) is both positive and a prime number; and with all the Pi(n) prime numbers.

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

External link

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