Friedlander–Iwaniec theorem

Friedlander–Iwaniec theorem   (Redirected from Bombieri–Friedlander–Iwaniec theorem) Jump to navigation Jump to search John Friedlander Henryk Iwaniec 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}} .

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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