Bouniakowsky Conjecture

Define a Bouniakowsky polynomial as an irreducible polynomial f(x) with integer coefficients, degree >1, and GCD(f(1),f(2),...)=1. The Bouniakowsky conjecture states that f(x) is prime for an infinite number of integers x (Bouniakowsky 1857). As an example of the greatest common divisor caveat, the polynomial 3x^2-x+2 is irreducible, but always divisible by 2.

Irreducible degree 1 polynomials (ax+b) always generate an infinite number of primes by Dirichlet's theorem. The existence of a Bouniakowsky polynomial that can produce an infinitude of primes is undetermined. The weaker fifth Hardy-Littlewood conjecture asserts that a^2+1 is prime for an infinite number of integers a>1.

Various prime-generating polynomials are known, but none of these always generates a prime (Legendre).

Worse yet, it is unknown if a general Bouniakowsky polynomial will always produce at least 1 prime number. For example, x^(12)+488669 produces no primes until x=616980, 764400, 933660, ... (OEIS A122131).

See also

Dirichlet's Theorem, Hardy-Littlewood Conjectures, Prime-Generating Polynomial

This entry contributed by Ed Pegg, Jr. (author's link)

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Bouniakowsky, V. "Nouveaux théorèmes relatifs à la distinction des nombres premiers et à la de composition des entiers en facteurs." Sc. Math. Phys. 6, 305-329, 1857.Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, pp. 332-333, 2005.Ruppert, W. M. "Reducibility of Polynomials f(x,y) Modulo p." 5 Aug 1998., N. J. A. Sequence A122131 in "The On-Line Encyclopedia of Integer Sequences."

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Bouniakowsky Conjecture

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Pegg, Ed Jr. "Bouniakowsky Conjecture." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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