Friedlander–Iwaniec theorem

The theorem was proved in 1997 by John Friedlander and Henryk Iwaniec.[1] Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.[2]

The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017.[3] In particular, they proved that the polynomial ${\displaystyle a^{2}+b^{4}}$ represents infinitely many primes when the variable ${\displaystyle b}$ is also required to be prime. Namely, if ${\displaystyle f(n)}$ is the prime numbers less than ${\displaystyle n}$ in the form ${\displaystyle a^{2}+b^{4},}$ then

${\displaystyle f(n)\sim v{\frac {x^{3/4}}{\log {x}}}}$

where

${\displaystyle v=2{\sqrt {\pi }}{\frac {\Gamma (5/4)}{\Gamma (7/4)}}\prod _{p\equiv 1{\bmod {4}}}{\frac {p-2}{p-1}}\prod _{p\equiv 3{\bmod {4}}}{\frac {p}{p-1}}.}$

When b = 1, the Friedlander–Iwaniec primes have the form ${\displaystyle a^{2}+1}$, forming the set

2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, … (sequence A002496 in the OEIS).

It is conjectured (one of Landau's problems) that this set is infinite. However, this is not implied by the Friedlander–Iwaniec theorem.

• Cipra, Barry Arthur (1998), "Sieving Prime Numbers From Thin Ore", Science, 279 (5347): 31, doi:10.1126/science.279.5347.31, S2CID 118322959.