New Fibonacci and Lucas primes

Authors:
Harvey Dubner and Wilfrid Keller

Journal:
Math. Comp. **68** (1999), 417-427

MSC (1991):
Primary 11A51; Secondary 11B39, 11--04

DOI:
https://doi.org/10.1090/S0025-5718-99-00981-3

Supplement:
Additional information related to this article.

MathSciNet review:
1484896

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers have been determined for and all probable prime Lucas numbers have been determined for . A rigorous proof of primality is given for and for numbers with , , , , , , , , the prime having 3020 digits. Primitive parts and of composite numbers and have also been tested for probable primality. Actual primality has been established for many of them, including 22 with more than 1000 digits. In a Supplement to the paper, factorizations of numbers and are given for as far as they have been completed, adding information to existing factor tables covering .

**1.**R. P. Brent,*MVFAC: A vectorized Fortran implementation of the elliptic curve method*, Comput. Sci. Lab., Austral. Nat. Univ., 1991.**2.**J. Brillhart, electronic mail to W. Keller dated 24 October 1994.**3.**-,*Note on Fibonacci primality testing*, Fibonacci Quart. (to appear).**4.**J. Brillhart, D. H. Lehmer, and J. L. Selfridge,*New primality criteria and factorizations of*, Math. Comp.**29**(1975), 620-647; Errata, Math. Comp.**39**(1982), 747.MR**52:5546**; MR**83j:10010****5.**J. Brillhart, P. L. Montgomery, and R. D. Silverman,*Tables of Fibonacci and Lucas Factorizations*, Math. Comp.**50**(1988), 251-260;*Supplement*, ibid., S1-S15. MR**89h:11002****6.**C. Caldwell,*Review of ``The Dubner PC cruncher - A microcomputer coprocessor card for doing integer arithmetic''*, J. Recreational Math.**25**(1993), 56-57.**7.**D. Jarden,*Recurring sequences*, 3rd ed., Riveon Lematematika, Jerusalem, 1973; Review of the 2nd ed. (by J. Brillhart) , Math. Comp.**23**(1969), 212-213; Errata, Math. Comp.**25**(1971), 200-201, and Math. Comp.**26**(1972), 1029-1030. MR**20:4663**(1st ed.); MR**53:4451**; MR**47:1731****8.**W. Keller,*Factors of and for*, machine-readable table, March 1996.**9.**-,*Some identities for primitive parts of Fibonacci and Lucas numbers*, unpublished notes, July 1996.**10.**P. L. Montgomery,*Status of composite Fibonacci and Lucas cofactors*, machine-readable table, August 1996.**11.**-,*Lucas extensions*, machine-readable table, August 1996.**12.**F. Morain,*On the primality of and*, preprint, July 1990.**13.**W. D. Neumann,*UBASIC: a Public-Domain BASIC for Mathematics*, Notices Amer. Math. Soc.**36**(1989), 557-559;*UBASIC Update*, ibid.**38**(1991), 196-197.

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Additional Information

**Harvey Dubner**

Affiliation:
449 Beverly Road, Ridgewood, New Jersey 07450

Email:
70327.1170@compuserve.com

**Wilfrid Keller**

Affiliation:
Regionales Rechenzentrum der Universität Hamburg, 20146 Hamburg, Germany

Email:
keller@rrz.uni-hamburg.de

DOI:
https://doi.org/10.1090/S0025-5718-99-00981-3

Keywords:
Fibonacci numbers,
Lucas numbers,
primality testing,
large primes,
prime primitive parts,
factor tables

Received by editor(s):
March 29, 1996

Received by editor(s) in revised form:
April 10, 1997

Article copyright:
© Copyright 1999
American Mathematical Society