Friedlander–Iwaniec theorem

In analytic number theory the Friedlander–Iwaniec theorem^[1] (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.^[2] 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^[3]) to the "Bounded gaps between primes"^[4] of Yitang Zhang.^[5] Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.^[6]

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.

References

1. van Golstein Brouwers, G.; Bamberg, D.; Cairns, J. (2004), "Totally Goldbach numbers and related conjectures" (PDF), Australian Mathematical Society Gazette 31 (4): 251–255 [p. 254].
2. 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.
3. Primes in Tuples I, D. A. Goldston, J. Pintz, C. Y. Yildirim, 2005. arXiv.org
4. Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics (Princeton University and the Institute for Advanced Study) 179: 1121–1174. doi:10.4007/annals.2014.179.3.7. Retrieved March 11, 2014. (subscription required)
5. Wilkinson, Alec. "The Pursuit of Beauty Yitang Zhang solves a pure-math mystery.". newyorker.com. Retrieved 1 February 2015. “Goldston-Pintz-Yıldırım”and “Bombieri-Friedlander-Iwaniec.” He [Yitang Zhang] said, “The first paper is on bound gaps, and the second is on the distribution of primes in arithmetic progressions. I compare these two together, plus my own innovations,
6. "Iwaniec, Sarnak, and Taylor Receive Ostrowski Prize"

Further reading

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