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Old 2010-08-08, 18:27   #1
wblipp's Avatar
May 2003
New Haven

3·787 Posts
Default Schinzel's Aurifeuillian style factorizations?

Are there other algebraic factorizations of cyclotomic numbers or polynomials that might be helpful to projects such as Cunningham, Homogenous Cunninghams, Mishima's Cyclotomic Numbers and Odd Perfect? I recently found this abstract which says in part
Aurifeuillian Factorization ... In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified all numbers that can be factored by such identities and we prove this if one accepts our definition
The Schinzel paper appears to be this one.. I haven't found a free discussion of this on the web, and I'm a few weeks away from being able to access this through a university. Does anybody here know about these additional factorizations?


Edit: Found the first paper on Granville's web site.

Last fiddled with by wblipp on 2010-08-08 at 18:43
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Old 2010-08-09, 11:04   #2
only_human's Avatar
"Gang aft agley"
Sep 2002

2·1,877 Posts
Binary BBP-Formulae for Logarithms and Generalized Gaussian-Mersenne Primes (2003) by Marc Chamberland notices some redundancies that crop up when developing BBP formulae that relate to Aurifeuillian identities.
This demonstrates why some calculations used in the last section to generate the list of primes were redundant. Indeed, in searching for various families of factors, similar identities arise. We now develop other Aurifeuillian identities, interesting for their own sake, and make connections to expressions used in the last section.
I hope this is useful. The relevant references in this paper are:
[5] R. Brent. Computing Aurifeuillian factors. Computational algebra and number theory (Sydney, 1992), Mathematics and its Applications, 325:201-212, Kluwer, Dordrecht, (1995).

[6] J. Brillhart, D.H. Lehmer, J.L. Selfridge, B. Tuckerman, and S.S. Wagsta,Jr. Factorizations of bn ± 1. American Mathematical Society, Providence, (1983).

[7] A. Schinzel. On Primitive Prime Factors of an - bn. Proceedings of the Cambridge Philosophical Society, 58(4):555-562, (1962).

[8] P. Stevenhagen. On Aurifeuillian Factorizations. Proceedings of the Konin-klijke Akademie van Wetenschappen, 90(4):451-468, (1987).

Last fiddled with by only_human on 2010-08-09 at 11:05
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Old 2010-08-15, 20:33   #3
maxal's Avatar
Feb 2005

22·32·7 Posts

Here is a couple of papers on Aurifeuillian factorizations:
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