Statement
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: # There are infinitely many positive integers such that all of are simultaneously prime numbers, or # There is an integer (called a ''fixed divisor'') which always divides the product . (Or, equivalently: There exists a prime such that for every there is an such that divides .) The second condition is satisfied by sets such as , since is always divisible by 2. It is easy to see that this condition prevents the ''first'' condition from being true. Schinzel's hypothesis essentially claims that condition 2 is the ''only'' way condition 1 can fail to hold. 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 successive values of . One can see by extrapolating with finite differences that this divisor will also divide all other values of too. Schinzel's hypothesis builds on the earlierExamples
As a simple example with , : 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 is often prime for up to 1500. As another example, take with and . The hypothesis then implies the existence of infinitely many twin primes, a basic and notorious open problem.Variants
As proved by Schinzel and Sierpiński in page 188 of it is equivalent to the following: if condition 2 does not hold, then there exists at least one positive integer such that all will be simultaneously prime, for any choice of irreducible integral polynomials 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 polynomials with integer coefficients, rather than integer-valued polynomials (such as , which takes integer values for all integer even though the coefficients are not integers).Previous results
The special case of a single linear polynomial isProspects and applications
The hypothesis is probably not accessible with current methods inExtension to include the Goldbach conjecture
The hypothesis doesn't cover Goldbach's conjecture, but a closely related version (hypothesis HN) does. That requires an extra polynomial , which in the Goldbach problem would just be , 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 : 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 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) that the polynomial : 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 all of its values as ''x'' runs over ''F''2 'u''are composite. Similar examples can be found with ''F''2 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.References
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{{Prime number conjectures Analytic number theory Conjectures about prime numbers Unsolved problems in number theory