Friedlander–Iwaniec theorem
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.
History
The theorem was proved in 1997 by John Friedlander and Henryk Iwaniec. Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.Refinements
The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017. 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 thenwhere
In 2024, a paper by Stanley Yao Xiao generalized the Friedlander—Iwaniec theorem and Heath-Brown—Li theorems to general binary quadratic forms, including indefinite forms. In particular one has, for a positive definite binary quadratic form satisfying, one has, for the prime indicator function and
and
with, the asymptotic formula:
Here is the discriminant of the quadratic form.
For indefinite, irreducible forms satisfying, put
Then one has the asymptotic formula
Special case
When, the Friedlander–Iwaniec primes have the form, forming the setIt is conjectured that this set is infinite. However, this is not implied by the Friedlander–Iwaniec theorem.