# Friedlander–Iwaniec theorem

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}} .

Contents 1 History 2 Refinements 3 Special case 4 References 5 Further reading 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] Refinements The theorem was refined by D.R. Heath-Brown and Xiannan Li in 2017.[3] In particular, they proved that the polynomial {displaystyle a^{2}+b^{4}} represents infinitely many primes when the variable {displaystyle b} is also required to be prime. Namely, if {displaystyle f(n)} is the prime numbers less then {displaystyle n} in the form {displaystyle a^{2}+b^{4},} then {displaystyle f(n)sim v{frac {x^{3/4}}{log {x}}}} where {displaystyle v=2{sqrt {pi }}{frac {Gamma (5/4)}{Gamma (7/4)}}prod _{pequiv 1{bmod {4}}}{frac {p-2}{p-1}}prod _{pequiv 3{bmod {4}}}{frac {p}{p-1}}.} Special case When b = 1, the Friedlander–Iwaniec primes have the form {displaystyle 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" ^ Heath-Brown, D.R.; Li, Xiannan (2017), "Prime values of {displaystyle a^{2}+p^{4}} ", Inventiones Mathematicae, 208: 441–499, doi:10.1007/s00222-016-0694-0. Further reading Cipra, Barry Arthur (1998), "Sieving Prime Numbers From Thin Ore", Science, 279 (5347): 31, doi:10.1126/science.279.5347.31, S2CID 118322959. Categories: Additive number theoryTheorems in analytic number theoryTheorems about prime numbers

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