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.