In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form . The first few such primes are
The difficulty in this statement lies in the very sparse nature of this sequence: the number of integers of the form less than is roughly of the order .
The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017.[3] In particular, they proved that the polynomial represents infinitely many primes when the variable is also required to be prime. Namely, if is the prime numbers less than in the form then
where
Special case
When b = 1, the Friedlander–Iwaniec primes have the form , forming the set