Tables of Fibonacci and Lucas factorizations

Authors:
John Brillhart, Peter L. Montgomery and Robert D. Silverman

Journal:
Math. Comp. **50** (1988), 251-260, S1

MSC:
Primary 11-04; Secondary 11B39

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917832-6

MathSciNet review:
917832

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We list the known prime factors of the Fibonacci numbers for and Lucas numbers for . We discuss the various methods used to obtain these factorizations, and primality tests, and give some history of the subject.

**[1]**J. Brillhart, "Fibonacci century mark reached,"*Fibonacci Quart.*, v. 1, 1963, p. 45.**[2]**J. Brillhart, "Some miscellaneous factorizations,"*Math. Comp.*, v. 17, 1963, pp. 447-450.**[3]**J. Brillhart, D. H. Lehmer & J. L. Selfridge, "New primality criteria and factorizations of ,"*Math. Comp.*, v. 29, 1975, pp. 620-647. MR**0384673 (52:5546)****[4]**J. Brillhart, "UMT of for and for ,"*Math. Comp.*, submitted to unpublished tables.**[5]**J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman & S. S. Wagstaff, Jr.,*Factorizations of*,*Up to High Powers*, Contemp. Math., vol. 22, Amer. Math. Soc., Providence, R.I., 1983. MR**715603 (84k:10005)****[6]**H. Cohen & H. W. Lenstra, Jr., "Primality testing and Jacobi sums,"*Math. Comp.*, v. 42, 1984, pp. 297-330. MR**726006 (86g:11078)****[7]**J. A. Davis & D. B. Holdridge,*Most Wanted Factorizations Using the Quadratic Sieve*, Sandia Report SAND84-1658, 1984.**[8]**J. L. Gerver, "Factoring large numbers with a quadratic sieve,"*Math. Comp.*, v. 41, 1983, pp. 287-294. MR**701639 (85c:11122)****[9]**D. Jarden,*Recurring Sequences*, 3rd ed., Riveon Lematematika, Jerusalem, 1973.**[10]**D. H. Lehmer, "An announcement concerning the Delay Line Sieve DLS 127,"*Math. Comp.*, v. 20, 1966, pp. 645-646.**[11]**H. W. Lenstra, Jr., "Factoring integers with elliptic curves," Ann. of Math. (To appear.)**[12]**Peter L. Montgomery, "Modular multiplication without trial division,"*Math. Comp.*, v. 44, 1985, pp. 519-521. MR**777282 (86e:11121)****[13]**Peter L. Montgomery, "Speeding the Pollard and elliptic curve methods of factorization,"*Math. Comp.*, v. 48, 1987, pp. 243-264. MR**866113 (88e:11130)****[14]**M. A. Morrison & J. Brillhart, "A method of factoring and the factorization of ,"*Math. Comp.*, v. 29, 1975, pp. 183-205. MR**0371800 (51:8017)****[15]**T. Naur,*Integer Factorization*, DAIMI PB-144, Computer Science Department, Aarhus University, Denmark, 1982.**[16]**T. Naur, "New integer factorizations,"*Math. Comp.*, v. 41, 1983, pp. 687-695. MR**717713 (85c:11123)****[17]**Robert D. Silverman, "The multiple polynomial quadratic sieve,"*Math. Comp.*, v. 48, 1987, pp. 329-339. MR**866119 (88c:11079)****[18]**Robert D. Silverman, "Parallel implementation of the quadratic sieve,"*The Journal of Supercomputing*, v. 1, 1987, no. 3.**[19]**H. C. Williams, " A method of factoring,"*Math. Comp.*, v. 39, 1982, pp. 225-234. MR**658227 (83h:10016)****[20]**H. C. Williams & J. S. Judd, "Some algorithms for prime testing using generalized Lehmer functions,"*Math. Comp.*, v. 30, 1976, pp. 867-886. MR**0414473 (54:2574)****[21]**H. C. Williams, private communication.

Retrieve articles in *Mathematics of Computation*
with MSC:
11-04,
11B39

Retrieve articles in all journals with MSC: 11-04, 11B39

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917832-6

Keywords:
Factor tables,
Fibonacci,
Lucas,
factorizations

Article copyright:
© Copyright 1988
American Mathematical Society