View source for Friedlander-Iwaniec theorem

The '''Friedlander-Iwaniec theorem''' asserts that there are infinitely many primes of the form <math>a^2+b^4</math>.  A more quantitative version of this theorem asserts, among other things, that there exists a k-digit prime of the form <math>a^2+b^4</math> for every k.

The relevance of this result to the [[finding primes]] project is that the set of k-digit numbers of the form <math>a^2+b^4</math> is relatively sparse, having cardinality <math>O( (10^k)^{3/4} )</math>, and is easy to enumerate.  This allows for a deterministic prime-searching algorithm with runtime <math>O( (10^k)^{3/4} )</math>.

Heath-Brown later showed a similar result for primes of the form <math>a^3+2b^3</math>, which leads to an algorithm of runtime  <math>O( (10^k)^{2/3} )</math>.  Related results for other binary cubic forms exist, see the survey of Moroz; but this is the sparsest set of polynomial forms currently known to capture primes.

# J. Friedlander, H. Iwaniec, [http://www.pnas.org/content/94/4/1054.abstract Using a parity-sensitive sieve to count prime values of a polynomial], PNAS 1997 94:1054-1058
# D.R. Heath-Brown, Primes represented by x3 +2y3; Acta Mathematica, 186 (2001), 1-84.
# B. Z. Moroz, [http://www.mpim-bonn.mpg.de/preprints/send?bid=3552 On the representation of primes by polynomials (a survey of some recent results)], Proceedings of the Mathematical Institute of the Belarussian Academy of Sciences, 13 (2005), no. 1, pp. 114-119.

== External links ==

* [http://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem The Wikipedia page for this theorem]