Theoretical Mathematics & Applications

Proof of Bunyakovsky's conjecture

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• Abstract

  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.

   

  Mathematics Subject Classification: 11A41, 11A51, 11B25, 11C08.

  Keywords: Bunyakovsky, Polynomials, Dirichlet, Arithmetic Progression, Prime.