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| 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
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| :''f''(''n''), ''g''(''n''), ...
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| with values at integers ''n'' of [[irreducible polynomial|irreducible]] [[integer-valued polynomial]]s
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| :''f''(''t''), ''g''(''t''), ...
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| 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, and on the [[Hardy–Littlewood conjecture]]s for multiple linear polynomials. It is in turn extended by the [[Bateman–Horn conjecture]].
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| ==Necessary limitations==
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| Such a conjecture must be subject to some [[necessary condition]]s. 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.
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| ==Fixed divisors pinned down==
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| 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
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| :''Q''(''x'')/''m''
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| is also an integer-valued polynomial. For example, we can say that
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| :(''x'' + 4)(''x'' + 7)
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| has 2 as fixed divisor. Such fixed divisors must be ruled out of
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| :''Q''(''x'') = Π ''f<sub>i</sub>''(''x'')
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| for any conjecture for polynomials ''f<sub>i</sub>'', ''i'' = 1 to ''k'', since their presence is quickly seen to contradict the possibility that ''f<sub>i</sub>''(''n'') can all be prime, with large values of ''n''.
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| ==Formulation of hypothesis H==
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| Therefore the standard form of '''hypothesis H''' is that if ''Q'' defined as above has ''no'' fixed prime divisor, then all ''f<sub>i</sub>''(''n'') will be simultaneously prime, infinitely often, for any choice of irreducible [[integral polynomial]]s ''f<sub>i</sub>''(''x'') with positive leading coefficients.
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| 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,
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| :''x''<sup>2</sup> + 1 | |
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| has no fixed prime divisor. We therefore expect that there are infinitely many primes
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| :''n''<sup>2</sup> + 1.
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| 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 ''n''<sup>2</sup>+1 is often prime for ''n'' up to 1500.
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| ==Prospects and applications==
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| The hypothesis is probably not accessible with current methods in [[analytic number theory]], but is now quite often used to prove [[conditional result]]s, 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. | |
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| ==Extension to include the Goldbach conjecture==
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| The hypothesis doesn't cover [[Goldbach's conjecture]], but a closely related version ('''hypothesis H<sub>N</sub>''') does. That requires an extra polynomial ''F''(''x''), which in the Goldbach problem would just be ''x'', for which
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| :''N'' − ''F''(''n'')
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| 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 | |
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| :''Q''(''n'')(''N'' − ''F''(''n''))
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| 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<sub>i</sub>''(''n'') prime numbers.
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| Not many cases of these conjectures are known; but there is a detailed quantitative theory ([[Bateman–Horn conjecture]]).
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| ==Local analysis==
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| 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
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| with no ''local obstruction'' to taking infinitely many prime values is conjectured to take infinitely many prime values. | |
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| ==An analogue that fails==
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| 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
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| ::<math>x^8 + u^3\,</math>
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| over the ring ''F''<sub>2</sub>[''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''<sub>2</sub>[''u''] are composite. Similar examples can be found with ''F''<sub>2</sub> 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. | |
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| ==References==
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| * {{cite book | title=Prime Numbers: A Computational Perspective | edition=2nd | first1=Richard | last1=Crandall | author1-link=Richard Crandall | first2=Carl B. | last2=Pomerance | author2-link=Carl Pomerance | publisher=[[Springer-Verlag]] | year=2005 | isbn=0-387-25282-7 | zbl=1088.11001 }}
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| * {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 }}
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| * {{cite book | last=Pollack | first=Paul | chapter=An explicit approach to hypothesis H for polynomials over a finite field | pages=259-273 | editor1-last=De Koninck | editor1-first=Jean-Marie | editor1-link=Jean-Marie De Koninck | editor2-last=Granville | editor2-first=Andrew | editor2-link=Andrew Granville | editor3-last=Luca | editor3-first=Florian | title=Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13--17, 2006 | location=Providence, RI | publisher=[[American Mathematical Society]] | series=CRM Proceedings and Lecture Notes | volume=46 | year=2008 | isbn=978-0-8218-4406-9 | zbl=1187.11046 }}
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| ==External links==
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| *[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]].
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| [[Category:Analytic number theory]]
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| [[Category:Conjectures about prime numbers]]
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