Friedlander–Iwaniec theorem

In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form $$a^2 + b^4$$. The first few such primes are


 * 2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, ….

The difficulty in this statement lies in the very sparse nature of this sequence: the number of integers of the form $$a^2+b^4$$ less than $$X$$ is roughly of the order $$X^{3/4}$$.

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 $$a^2 + b^4$$ represents infinitely many primes when the variable $$b$$ is also required to be prime. Namely, if $$f(n)$$ is the prime numbers less than $$n$$ in the form $$a^2 + b^4,$$ then

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

where

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

Special case
When $b = 1$, the Friedlander–Iwaniec primes have the form $$a^2+1$$, 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, ….

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