Dickson's conjecture

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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.

Weaker facts and conjectures