Friedlander-Iwaniec theorem
The Friedlander-Iwaniec theorem asserts that there are infinitely many primes of the form [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ a^2+b^4 }[/math] is relatively sparse, having cardinality [math]\displaystyle{ O( (10^k)^{3/4} ) }[/math], and is easy to enumerate. This allows for a deterministic prime-searching algorithm with runtime [math]\displaystyle{ O( (10^k)^{3/4} ) }[/math].
Heath-Brown later showed a similar result for primes of the form [math]\displaystyle{ a^3+2b^3 }[/math], which leads to an algorithm of runtime [math]\displaystyle{ 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, 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, [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.
