Bunyakovsky conjecture

The Bunyakovsky conjecture (or Bouniakowsky conjecture) stated in 1857 by the Ukrainian mathematician Viktor Bunyakovsky, claims that an irreducible polynomial of degree two or higher with integer coefficients generates for natural arguments either an infinite set of numbers with greatest common divisor exceeding unity, or infinitely many prime numbers.

An example is provided by the polynomial "f"("x") = "x"² + 1, for which some of the prime numbers generated are listed below:

"x" "x"² + 1 -------------- 1 2 2 5 4 17 6 37 10 101 14 197 16 257 20 401 24 577 26 677 36 1297 The fifth Hardy-Littlewood conjecture—a special case of the Bunyakovsky conjecture—states that $x^2+1$ generates infinitely many prime values for integer "x" > 1. To date, the Bunyakovsky conjecture has not been proven correct, nor is a counterexample known.

The Bunyakovsky conjecture can be seen as an extension of Dirichlet's theorem, which states that irreducible degree one polynomials always generate an infinite number of primes.

ee also

*Integer-valued polynomial

References

*MathWorld|urlname=BouniakowskyConjecture|title=Bouniakowsky conjecture|author=Ed Pegg, Jr.
*cite journal|last = Rupert|first = Wolfgang M.|title = Reducibility of polynomials "f"("x", "y") modulo "p"|journal = Arxiv.org|date = 1998-08-05|url = http://arxiv.org/pdf/math/9808021
*cite journal|last = Bouniakowsky|first = V.|title = Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la décomposition des entiers en facteurs|journal = Mém. Acad. Sc. St. Pétersbourg|volume = 6|pages = 305–329|date = 1857