Friedlander–Iwaniec theorem

In analytic number theory the Friedlander–Iwaniec theorem (or Bombieri–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, … (sequence A028916 in 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. It uses sieve techniques, in a form which extends Enrico Bombieri's asymptotic sieve. Friedlander–Iwaniec theorem is one of the two keys (the other is the 2005 work of Goldston-Pintz-Yıldırım) to the "Bounded gaps between primes" of Yitang Zhang. Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.

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, … (sequence A002496 in OEIS).

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

Further reading

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