Friedlander–Iwaniec theorem

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In analytic number theory the Friedlander–Iwaniec theorem[1] asserts 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}.

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. Iwaniec was awarded the 2001 Ostrowski Prize in part for his contributions to this work.[3]

This result, however, does not imply that there are an infinite number of primes of form a^2+1, or

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).

as the latter is still an unsolved problem (one of Landau's problems).

References[edit]

  1. ^ van Golstein Brouwers, G.; Bamberg, D.; Cairns, J. (2004), "Totally Goldbach numbers and related conjectures", 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. ^ "Iwaniec, Sarnak, and Taylor Receive Ostrowski Prize"

Further reading[edit]