Schinzel's hypothesis H is one of the most famous open problems in number theory, a very broad generalization of widely open conjectures such as the twin prime conjecture. It is named after Andrzej Schinzel. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementFor any finite collection of nonconstant irreducible polynomials with integer coefficients and positive leading coefficients, the hypothesis claims that one of two things holds: either infinitely many positive integers make every polynomial in the collection simultaneously prime, or there is some fixed integer greater than one that always divides the product of the polynomials' values, ruling out primality for every input. 1 Progress Toward ResolutionThe only case actually proved is a single linear polynomial, which is Dirichlet's theorem on arithmetic progressions; no case of degree greater than one, and no system of more than one polynomial, is known to hold. Chen's theorem and a result of Iwaniec give partial almost-prime approximations, and Skorobogatov and Sofos proved the hypothesis for almost all polynomials of any fixed degree, though not for any specific one beyond the linear case. 1 Classification
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Source Schinzel's Hypothesis H (Wikipedia)
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1. Schinzel's Hypothesis H (Wikipedia)
Wikimedia FoundationLead section
Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture
Statement section, opening sentence
The hypothesis claims that for every finite collection {f1, f2, ..., fk} of nonconstant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds:
Previous results section
In fact, this special case is the only known instance of Schinzel's Hypothesis H. We do not know the hypothesis to hold for any given polynomial of degree greater than 1, nor for any system of more than one polynomial.
Lead section, opening sentence
Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory.
In Branch: Number Theory, Lead sentence
of the most famous open problems in the topic of number theory.
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