In 1857, twenty years
after Dirichlet's theorem on arithmetic progressions, the conjecture of the
Ukrainian mathematician Victor Y. Bunyakovsky (1804-1889) is already a try to
generalize this theorem to polynomial integer functions of degree m>1. This
conjecture states that under three conditions a polynomial integer function of
degree m>1 generates infinitely many primes.
The main contribution
of this paper is to introduce a new approach to this conjecture. The key ideas
of this new approach is to relate the conjecture to a general theory (here
arithmetic progressions) and use the active constraint of this theory
(Dirichlet's theorem) to achieve the proof.
Subject Classification: 11A41,
11A51, 11B25, 11C08.
Dirichlet, Arithmetic Progression, Prime.