Friedlander–Iwaniec theorem

In analytic number theory the Friedlander–Iwaniec theorem states that there are infinitely many prime numbers of the form ${\displaystyle a^{2}+b^{4}}$. 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, … (sequence A028916 in the OEIS).

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