Aurifeuillian factorization

Granville, Andrew and Pleasants, Peter (2006) Aurifeuillian factorization. Mathematics of Computation, 75 253: 497-508. doi:10.1090/S0025-5718-05-01766-7

Author Granville, Andrew
Pleasants, Peter
Title Aurifeuillian factorization
Journal name Mathematics of Computation   Check publisher's open access policy
ISSN 0025-5718
Publication date 2006-01
Sub-type Article (original research)
DOI 10.1090/S0025-5718-05-01766-7
Volume 75
Issue 253
Start page 497
End page 508
Total pages 12
Editor Susanne C Brenner
Ronald F. A .Cools
Harold Niederreiter
Place of publication Providence, RI, United States
Publisher American Mathematical Society
Collection year 2005
Language eng
Subject 230102 Number Theory And Field Theory
780101 Mathematical sciences
Abstract The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing bn by a polynomial in such a way that polynomial factorization is possible. For example, by substituting y = 2k into the polynomial factorization (2y2)2+1 = (2y2−2y+1)(2y2+2y+1) we can partially factor 24k+2+1. 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 of what “such an identity” is. We then develop our theme to similarly factor f(bn) for any given polynomial f, using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: 2006 Higher Education Research Data Collection
School of Physical Sciences Publications
Version Filter Type
Citation counts: TR Web of Science Citation Count  Cited 0 times in Thomson Reuters Web of Science Article
Scopus Citation Count Cited 1 times in Scopus Article | Citations
Google Scholar Search Google Scholar
Created: Wed, 15 Aug 2007, 07:44:31 EST