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 {\displaystyle a^{2}+b^{4}} 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 \( a^2+b^4 \) less than X is roughly of the order \( X^{3/4} \).

History

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

When b = 1, the Friedlander–Iwaniec primes have the form a \( a^2+1 \), 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.

References

Friedlander, John; Iwaniec, Henryk (1997), "Using a parity-sensitive sieve to count prime values of a polynomial", PNAS, 94 (4): 1054–1058, doi:10.1073/pnas.94.4.1054, PMC 19742, PMID 11038598.

"Iwaniec, Sarnak, and Taylor Receive Ostrowski Prize"