Schinzel

Schinzel's Hypothesis H is a far-reaching conjecture in number theory formulated by Polish mathematician Andrzej Schinzel. It generalizes several known theorems and conjectures about prime numbers and polynomials.

The hypothesis, in its simplest form, deals with a finite set of irreducible polynomials f₁(x), f₂(x), ..., f_k(x) with integer coefficients, where the leading coefficient of each polynomial is positive, and with the condition that they satisfy the Schinzel condition. The Schinzel condition states that there is no integer n for which all of the polynomials f₁(n), f₂(n), ..., f_k(n) are divisible by some fixed prime p. In other words, for every prime p, there exists an integer n such that at least one of the values f_i(n) is not divisible by p.

Schinzel's Hypothesis H then states that under these conditions, there exist infinitely many positive integers n such that f₁(n), f₂(n), ..., f_k(n) are all prime numbers.

The hypothesis has several notable corollaries. For instance, taking k = 1 and f₁(x) = x, it implies that there are infinitely many prime numbers. Taking k = 1 and f₁(x) = x² + 1, it implies that there are infinitely many prime numbers of the form n² + 1, a famous unsolved problem. Setting k = 2, f₁(x) = x, and f₂(x) = x + 2, it implies the twin prime conjecture.

Schinzel's Hypothesis H is considered one of the most important unsolved problems in number theory due to its generality and the fact that it encompasses many other famous conjectures about prime numbers. Its truth would provide a unified framework for understanding the distribution of prime values of polynomials. It is a vast generalization of the Bunyakovsky conjecture.