The Bunyakovsky conjecture gives a criterion for a one-variable polynomial with integer coefficients to yield infinitely many prime values. It was stated in 1857 by Russian mathematician Viktor Bunyakovsky, who held that if the polynomial's leading coefficient is positive, the polynomial is irreducible over the integers, and its values share no common factor greater than 1, then it is prime for infinitely many positive integers. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementLet f(x) be a one-variable polynomial with integer coefficients whose leading coefficient is positive, which is irreducible over the integers, and whose values f(1), f(2), f(3), ... have no common factor larger than 1. Then f(n) is prime for infinitely many positive integers n. 1 Proposed Year Progress Toward ResolutionOpen. The only proved case is polynomials of degree 1, which is Dirichlet's theorem on primes in arithmetic progressions; no case of degree greater than 1 is proved, although numerical evidence in higher degree is consistent with the conjecture. 1 Classification
Resolution Status Prize Status
Prize Status (category) Sources
1. Bunyakovsky Conjecture (Wikipedia)
Wikimedia FoundationLead section
It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky
Statement section, sufficiency sentence
then f(n) is prime for infinitely many positive integers n
Discussion section, higher degree sentence
No single case of Bunyakovsky's conjecture for degree greater than 1 is proved, although numerical evidence in higher degree is consistent with the conjecture.
Discussion section, degree 1 sentence
To date, the only case of Bunyakovsky's conjecture that has been proved is that of polynomials of degree 1.
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