Bunyakovsky conjecture

The Bunyakovsky conjecture states that every irreducible integer polynomial with positive leading coefficient and no fixed prime divisor takes prime values infinitely often.

Irreducible polynomial: A nonconstant polynomial that cannot be factored into lower-degree polynomials over a specified coefficient field or ring. Irreducibility is one of the conjecture’s explicit hypotheses.
Admissible polynomial: An integer polynomial whose values are not all divisible by any one prime. This is the standard local condition encoded by having no fixed prime divisor.
Linear polynomial: A degree-one polynomial of the form ax+b. For positive slope and coprime coefficients, its prime-value claim follows from Dirichlet’s theorem.
Viktor Bunyakovsky: A nineteenth-century Russian mathematician known for work in number theory, probability, and inequalities. He formulated the conjecture in 1857.
Bunyakovsky conjecture for quadratic polynomials: The claim that every admissible irreducible quadratic integer polynomial with positive leading coefficient takes prime values infinitely often. Even this restricted form remains open, including for n²+1.
Fixed divisor: The greatest common divisor of all values taken by an integer polynomial at integer inputs. A fixed divisor greater than one prevents every polynomial value from being prime.
Local obstruction: A congruence or divisibility condition that rules out a proposed integer solution or value. A fixed prime divisor is a local obstruction to prime values.
Dirichlet's theorem on arithmetic progressions: A theorem stating that every arithmetic progression with coprime initial term and common difference contains infinitely many primes. It proves the Bunyakovsky prediction for linear polynomials.
Legendre's constant: A proposed constant term in a nineteenth-century formula approximating the nth prime. It belongs to the same historical effort to describe the distribution of prime numbers.
Twin prime conjecture: The conjecture that infinitely many pairs of primes differ by two. It is another unproved prediction about prime patterns, corresponding to simultaneous linear polynomial values.

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