In mathematics, 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. The hypothesis is named after Andrzej Schinzel.
Statement The hypothesis claims that for every finite collection { f 1 , f 2 , … , f k } {\displaystyle \{f_{1},f_{2},\ldots ,f_{k}\}} of nonconstant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds:
There are infinitely many positive integers n {\displaystyle n} such that all of f 1 ( n ) , f 2 ( n ) , … , f k ( n ) {\displaystyle f_{1}(n),f_{2}(n),\ldots ,f_{k}(n)} are simultaneously prime numbers, or There is an integer m > 1 {\displaystyle m>1} (called a "fixed divisor"), which depends on the polynomials, which always divides the product f 1 ( n ) f 2 ( n ) ⋯ f k ( n ) {\displaystyle f_{1}(n)f_{2}(n)\cdots f_{k}(n)} . (Or, equivalently: There exists a prime p {\displaystyle p} such that for every n {\displaystyle n} there is an i {\displaystyle i} such that p {\displaystyle p} divides f i ( n ) {\displaystyle f_{i}(n)} .) The second condition is satisfied by sets such as f 1 ( x ) = x + 4 , f 2 ( x ) = x + 7 {\displaystyle f_{1}(x)=x+4,f_{2}(x)=x+7} , since ( x + 4 ) ( x + 7 ) {\displaystyle (x+4)(x+7)} 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: letting Q ( x ) = f 1 ( x ) f 2 ( x ) ⋯ f k ( x ) {\displaystyle Q(x)=f_{1}(x)f_{2}(x)\cdots f_{k}(x)} , compute the greatest common divisor of deg ( Q ) + 1 {\displaystyle \deg(Q)+1} successive values of Q ( n ) {\displaystyle Q(n)} . One can see by extrapolating with finite differences that this divisor will also divide all other values of Q ( n ) {\displaystyle Q(n)} too. Schinzel's hypothesis builds on the earlier Bunyakovsky conjecture, for a single polynomial, and on the Hardy–Littlewood conjectures and Dickson's conjecture for multiple linear polynomials. It is in turn extended by the Bateman–Horn conjecture.
Examples As a simple example with k = 1 {\displaystyle k=1} ,
x 2 + 1 {\displaystyle x^{2}+1}
has no fixed prime divisor. We therefore expect that there are infinitely many primes
n 2 + 1 {\displaystyle n^{2}+1} . 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 2 + 1 {\displaystyle n^{2}+1} is often prime for n {\displaystyle n} up to 1500. As another example, take k = 2 {\displaystyle k=2} with f 1 ( x ) = x {\displaystyle f_{1}(x)=x} and f 2 ( x ) = x + 2 {\displaystyle f_{2}(x)=x+2} . The hypothesis then implies the existence of infinitely many twin primes, a famous open problem.
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