# Dickson's conjecture

From Number

## Contents

## Statement

Suppose are integers with all the . Then, consider the polynomials:

Then, one of the following is true:

- There is a prime number such that the product is times an integer-valued polynomial. In other words, one of the polynomials is always congruent to 1 modulo .
- There exist infinitely many [[natural number]s for which
*all*the values are*simultaneously*prime.

## Related facts and conjectures

### Stronger facts and conjectures

- Schinzel's hypothesis H generalizes from linear polynomials to polynomial of arbitrary degree.
- Bateman-Horn conjecture further generalies Schinzel's hypothesis H by providing an asymptotic quantitative estimate of the frequency of occurrence of primes.