## New Fibonacci and Lucas primes (1999)

Venue: | Math. Comp |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Dubner99newfibonacci,

author = {Harvey Dubner and Wilfrid Keller},

title = {New Fibonacci and Lucas primes},

journal = {Math. Comp},

year = {1999},

volume = {68},

pages = {417--427}

}

### OpenURL

### Abstract

Abstract. Extending previous searches for prime Fibonacci and Lucas numbers, all probable prime Fibonacci numbers Fn have been determined for 6000 <n≤50000 and all probable prime Lucas numbers Ln have been determined for 1000 <n≤50000. A rigorous proof of primality is given for F9311

### Citations

14 | The Dubner PC Cruncher { a microcomputer coprocessor card for doing integer arithmetic, review in - Caldwell - 1993 |

9 |
New primality criteria and factorizations of 2 m 1
- Brillhart, Lehmer, et al.
- 1975
(Show Context)
Citation Context ... provision of a completely factored part of N −1 orofN+1that exceeds in magnitude N 1/2 or lies, at least, between N 1/3 and N 1/2 . We first state the theorems, which are derived from those found in =-=[4]-=-, and then we discuss their application from a practical point of view. Let N − 1=G·H,whereGisacompletely factored portion of N − 1, H>1, and (G, H) =1. Theorem 1. Suppose G2 >N. If for each prime pi ... |

1 |
MVFAC: A vectorized Fortran implementation of the elliptic curve method
- Brent
- 1991
(Show Context)
Citation Context ...rs was often decisive for the completion of a proof. The means used were essentially the factoring and prime-proving procedures of the UBASIC package [13], R. P. Brent’s vectorized ECM implementation =-=[1]-=-, and the first author’s program for the “p − 1” method. Based on a rather modest collection of factorizations of numbers Ln we had gathered for 500 <n≤1000, P. Montgomery has added to this a consider... |

1 |
electronic mail to W. Keller dated 24
- Brillhart
- 1994
(Show Context)
Citation Context ...7 ∗ , 35999 ∗ , 37511 ∗ ,andfornoothern≤50000. The interval n ≤ 1000 had been covered by Brillhart; cf. the review of [7]. Williams searched 1000 <n≤6000 for probable primes (as reported by Brillhart =-=[2]-=-) and showed that F2971 was indeed a prime, while F4723 and F5387 were subsequently proven prime by Morain [12] using techniques similar to those we will be describing below. Also, Ln has been shown t... |

1 |
Recurring sequences, 3rd ed., Riveon Lematematika
- Jarden
- 1973
(Show Context)
Citation Context ...s numbers Ln are defined recursively by the formulas Fn+2 = Fn+1 + Fn, n ≥ 0, F0 =0, F1 =1, Ln+2 = Ln+1 + Ln, n ≥ 0, L0 =2, L1 =1. These numbers have many interesting properties and applications; see =-=[7]-=- and the historical references therein. Here we report on a search for new primes Fn and Ln which extends previous work of J. Brillhart, H. C. Williams, and F. Morain. It turned out that Fn is a prime... |

1 |
Factors of Fn and Ln for 1000 machine-readable table
- Keller
- 1996
(Show Context)
Citation Context ...omplete the proof for L∗10821 . Regardless of their possible involvement in primality proofs, we have continued doing factoring work for numbers Fn and Ln with 1000 <n≤9750. The result is recorded in =-=[8]-=- and includes, in particular, a listing of all primitive prime divisors p<max(234 , 4 · 106n). They were determined by trial division taking advantage of certain linear dependencies on n that are summ... |

1 |
Status of composite Fibonacci and Lucas cofactors, machine-readable table
- Montgomery
- 1996
(Show Context)
Citation Context ...or a prime-proof in those cases had only been determined experimentally. 8. Factor tables Many of the factorizations needed for our primality proofs were taken from the tables in [5] and their update =-=[10]-=-, which cover Fibonacci numbers Fn for odd n ≤ 1000 and Lucas numbers Ln for all n ≤ 500. However, many factorizations needed beyond these limits were specifically obtained during the course of this i... |

1 |
extensions, machine-readable table
- Lucas
- 1996
(Show Context)
Citation Context ...factorizations of numbers Ln we had gathered for 500 <n≤1000, P. Montgomery has added to this a considerable number of more significant factorizations. Currently he is maintaining the extension table =-=[11]-=- covering that segment. Special mention should be made of two “difficult” factorizations in the extended range that were kindly produced at our request. H. J. J. te Riele, using PPMPQS, split the 90-d... |

1 |
On the primality of F4723 and F5387
- Morain
- 1990
(Show Context)
Citation Context ...view of [7]. Williams searched 1000 <n≤6000 for probable primes (as reported by Brillhart [2]) and showed that F2971 was indeed a prime, while F4723 and F5387 were subsequently proven prime by Morain =-=[12]-=- using techniques similar to those we will be describing below. Also, Ln has been shown to be a prime (or a probable prime) for n =0,2,4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 1... |

1 |
UBASIC: a Public-Domain BASIC for Mathematics, Notices
- Neumann
- 1989
(Show Context)
Citation Context ...ncise treatment, but were of a size just accessible to a general prime-proving procedure, have been subjected to APRT-CL, the Cohen-Lenstra version of the Adleman-Pomerance-Rumely test implemented in =-=[13]-=-. The largest prime primitive part confirmed in this way was the 843-digit number F ∗ 7917, which consumed 134 hours and 28 minutes on a Pentium 100 processor.s424 HARVEY DUBNER AND WILFRID KELLER Tab... |