# Dickson's conjecture

## 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.