The Bouniakowsky conjecture states that an irreducible polynomial 
with integer coefficients,
positive leading coefficient, and 
takes prime
values for infinitely many positive integers 
(Bouniakowsky 1857). Irreducibility is over the rational
numbers. A polynomial satisfying these conditions
and having degree greater than 1 is called a
Bouniakowsky polynomial here. The greatest
common divisor condition excludes the irreducible
polynomial 
,
which is divisible by 2 for every integer 
.
For a linear polynomial 
, the corresponding hypotheses are 
and 
. Under these conditions, Dirichlet's
theorem gives infinitely many primes. No polynomial
with integer coefficients
and degree greater than 1 is known to take infinitely
many prime values at positive
integer inputs (Jones and Zvonkin 2023, Ismailescu and Lee 2025). The assertion
that 
is prime for infinitely many integers 
is a special case and one of Landau's problems.
No nonconstant polynomial with integer coefficients can take prime
values at every positive integer input. If 
is prime at a positive
integer 
,
then 
divides 
for every positive integer 
, while 
eventually exceeds 
.
It is also unknown whether every Bouniakowsky polynomial takes even one prime value at a positive integer input. This assertion
for every such polynomial is equivalent to the full
conjecture, since applying it to 
for each positive integer 
would give prime values at arbitrarily large inputs
(Jones and Zvonkin 2023, p. 3). For example, 
first takes prime
values at 
,
764400, 933660, ... (Ismailescu and Lee 2025, p. 2; OEIS A122131).
Modulo 
." 5 Aug 1998.