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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Three new factors of Fermat numbers


Authors: R. P. Brent, R. E. Crandall, K. Dilcher and C. Van Halewyn
Journal: Math. Comp. 69 (2000), 1297-1304
MSC (1991): Primary 11Y05, 11B83, 11Y55; Secondary 11-04, 11A51, 11Y11, 11Y16, 14H52, 65Y10, 68Q25
Published electronically: March 1, 2000
MathSciNet review: 1697645
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Abstract | References | Similar Articles | Additional Information

Abstract:

We report the discovery of a new factor for each of the Fermat numbers $F_{13}, F_{15}, F_{16}$. These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with $2391$, $9808$ and $19694$ decimal digits respectively.


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Additional Information

R. P. Brent
Affiliation: Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, UK
Email: Richard.Brent@comlab.ox.ac.uk

R. E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, OR 97202, USA
Email: crandall@reed.edu

K. Dilcher
Affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada
Email: dilcher@cs.dal.ca

C. Van Halewyn
Affiliation: Department of Computer Science and Engineering, Oregon Graduate Institute
Address at time of publication: Deutsche Bank AG, London, England
Email: Christopher.van-halewyn@db.com

DOI: http://dx.doi.org/10.1090/S0025-5718-00-01207-2
PII: S 0025-5718(00)01207-2
Keywords: Discrete weighted transform, DWT, ECM, elliptic curve method, factorization, Fermat number, $F_{13}$, $F_{15}$, $F_{16}$, integer factorization
Received by editor(s): July 29, 1997
Published electronically: March 1, 2000
Article copyright: © Copyright 2000 American Mathematical Society