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MATHEMATICS OF COMPUTATIONVOLUME 50, NUMBER 181JANUARY 1988, PAGES 251-260

Tables of Fibonacci and Lucas Factorizations

By John Brillhart, Peter L. Montgomery, and Robert D. SilvermanDedicated to Dov Jarden

Abstract. We list the known prime factors of the Fibonacci numbers Fn for n < 999and Lucas numbers Ln for n < 500. We discuss the various methods used to obtainthese factorizations, and primality tests, and give some history of the subject.

1. Introduction. In the Supplements section at the end of this issue we give intwo tables the known prime factors of the Fibonacci numbers Fn, 3 < n < 999, nodd, and the Lucas numbers Ln, 2 < n < 500. The sequences Fn and Ln aredefined recursively by the formulas

. . ^n+2 = Fn+X + Fn, Fo = 0, Fi = 1,

Ln+2 = Ln+i + Ln, in = 2, L\ = 1.

The use of a different subscripting destroys the divisibility properties of thesenumbers.

We also have the formulasan - 3a

(1-2) Fn = --£-, Ln = an + ßn,a — ß

where a = (1 + \/B)/2 and ß — (1 - v^5)/2. This paper is concerned with themultiplicative structure of Fn and Ln. It includes both theoretical and numericalresults.

2. Multiplicative Structure of Fn and Ln. The identity

(2.1) F2n = FnLn

follows directly from (1.2). Although the Fibonacci and Lucas numbers are definedadditively, this is one of many multiplicative identities relating these sequences.The identities in this paper are derived from the familiar polynomial factorization

(2.2) xn-yn = ]\*d(x,y), n>\,d\n

where $d(x,y) is the dth cyclotomic polynomial in homogeneous form.Define the primitive part F¿ of F¿ to be

(2.3) /■; = {'•'_____ ' l*d(o,«, d>2.

Received February 9, 1987; revised April 27, 1987.1980 Mathematics Subject Classification (1985 Revision). Primary 11A51, 11B39; Secondary

68-04.Key words and phrases. Factor tables, Fibonacci, Lucas, factorizations.

©1988 American Mathematical Society0025-5718/88 $1.00 + $.25 per page

251

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252 JOHN BRILLHART, PETER L. MONTGOMERY, AND ROBERT D. SILVERMAN

Then we have the factorization

(2.4) Fn = Y[F*d, n>\.d\n

Here the F¿ are rational integers, computable by the inverse formula

(2.5) Fi = X\F»m\ d>l,6\d

where p is the Möbius function. The ratio Fn = Fn/F* is called the algebraic partofFn.

Formula (2.4) reduces factoring Fn to factoring the FJ's. Formula (2.5) showsthat the primitive part can be obtained without factoring.

A prime factor of F„ (resp. Ln) is called primitive if it does not divide Fk (resp.Lk) for 1 < k < n; otherwise it is called algebraic. A composite factor of Fn isalso called algebraic if it is a product of prime algebraic factors. Any prime divisorof F¿ (resp. L'n) is necessarily algebraic, but under certain circumstances a primedivisor of F* (resp. L*n) is not primitive. Such an algebraic prime factor p of F*(resp. L* ) is called intrinsic and is listed as p* in these tables. This occurs exactlywhen n = prm, r > 1, where p is a primitive factor of Fm (resp. Lm). In this casep always divides F* (resp. L* ) to just the first power.

Example. The factorization of F105, given by (2.4), is

j? _ 1 f jr>* _ jp* jp* ¡p* jp* jp* jp* jp* jp*^105- [I I'd -•i,l-f3^5-f7^15^21-I'35-f105-d|105

This factorization is abbreviated in Table 2 as

105 (3,5,7,15,21,35) 8288823481.

Here the numbers within the parentheses are the subscripts of the algebraicfactors FJ, 1 < d < 105. (The factor F* = 1 is omitted.) The primitive part FX05 —8288823481 is given after the parentheses. The lines in Table 2 corresponding tothe numbers inside the parentheses are:

32557 13

15 (3,5)6121 (3.7)42135 (5,7) 141961

The factorization of Fins is then obtained by collecting the primitive primefactors from their respective lines. These follow the parentheses (if any) on theseven lines and are underlined above for emphasis. Thus,

F105 = 2 • 5 • 13 • 61 • 421 • 141961 • 8288823481.

Because of (2.1), the algebraic multiplicative structure for Ln can be deriveddirectly from that of F2n. Let n = 2sm, where m is odd. Then

(2.6) Ln = n¿2M, «>1,d\m


TABLES OF FIBONACCI AND LUCAS FACTORIZATIONS 253

where

(2.7) L;,d = F;,+ld = YlL^d/6\ d>l.S\d

The primitive part of Ln is L*n — F2*n. The algebraic part of Ln is

(2-8) L'n = Ln/L*n.

Furthermore, as a result of a generalization by Lucas of a special identity discoveredby Aurifeuille, we also have for odd n

i^n = a +p = a4n _ a3nßn + a2n/32n _ anß3n + /?4nLn an + ßn

= (a2n - 3anßn + ß2n)2 + 5anßn(an - ßn)2

= (5F2 + l)2 - 25F2

= (hF2 + 5Fn + 1)(5F2 - 5Fn + 1) (using aß = -1 and a - ß = VE).Consequently, we have the special Aurifeuillian factorization

(2.9) L5n = LnA5nB5n, n odd,

whereA5„ = 5Fn2 - 5Fn + 1, B5n = 5F2 + 5Fn + 1.

This decomposition means that these ¿5n's have two different algebraic factoriza-tions. For example, from (2.6) and (2.9)

¿ios = il Ld = LXL3L5L7LX5L2XL35L105d|105

and¿105 = ¿2lAi05.Bi05-

Primitive parts An and ß* can also be defined for An and Bn. Let n > 1 be oddand set n = 5"m,s > 0,5 f ra. Let £d — \ (l + (§)), where (g) is the Legendresymbol. Let

(2-10) ÎÎBin=\[[(AsnldY-*


Since j4g = A\5 = 1, these are omitted in Table 3, while Bg" is written as Lg andBU as L*X5.

Those Lucas numbers which do not have an Aurifeuillian factorization appearin the tables in the same format as the Fibonacci factorizations. However, theAurifeuillian factorizations appear in an expanded format. For example, the abovefactorization appears as:

105 (3,7,21) ABA (15,35B) 21211B (5,35A) 767131.

The list of numbers immediately after the index 105 indicate that L105 has thealgebraic factors L\,L7, and L\x. Furthermore, A\QX> has algebraic factors L*X5 andB35, while Biog has algebraic factors L*b and A35. In computing A* and B*, thefollowing result is sometimes useful [9, p. 16]:

Theorem 1 (Crossover Theorem). For oddk,n> l where (5,fc) = l and(|) is the Jacobi symbol,

if ( 7 j = !> then A5n I A5kn and B5n | B5fcn;

if ( 7 J = -1, then A5n | B5kn and B5n | A5fcn.

The tables are organized using formulas (2.4) and (2.6). As a result, no primefactor appears explicitly more than once in the tables (except intrinsic factors andthe repeated factor 2 of L3). Where space permits, we list the known factors in theirentirety on a single line. We list all prime factors of 25 digits or less, carrying overto a second line, without breaking the factor, when necessary. All other factors arelisted as either Pxx or Cxx, indicating respectively a prime or a composite cofactorof xx digits. When a factorization is incomplete, we leave space on the line for newfactors to be inserted by hand.

3. Factorization Methods. A variety of methods have been used to effect thefactorizations given herein. These include the Pollard p - 1 and Brent-Pollard Rhomethods [13], the analogous p+ 1 method [19], the Continued Fraction (CFRAC)method of Morrison and Brillhart [14], Pomerance's Quadratic Sieve (QS) method[8], along with its extensions and improvements (MP-QS) [17], [18], and Lenstra'sElliptic Curve Method (ECM) [11], [13]. Of course, many of the smaller primefactors are quite old, and were originally found by trial division or the difference ofsquares method.

Some of the methods utilize the form of the prime divisors given by the followingtheorems [9, p. 11].

THEOREM 2. Let n be odd and let p be an odd, primitive prime divisor of Fn.Then

(i) p = 1 mod 4.(ii) if p = ±1 mod 10, then p = 1 mod 4n.(iii) if p = ±3 mod 10, then p = 2n - 1 mod 4n.



THEOREM 3. Let n be positive and let p be an odd, primitive prime divisor ofLn. Then

(i) ifp = ±l mod 10, then p = 1 mod 2n.(ii) if p = ±3 mod 10, then p = — 1 mod 2n.

4. Primality Testing. In [9, p. 36], Brillhart gave the following results of pri-mality tests on the Fibonacci and Lucas numbers: Fn, 3 < n < 1000, is prime if andonly if n = 3,4,5,7,11,13,17,23,29,43,47,83,131,137,359,431,433,449,509,569,571; L„, 0 < n < 500, is prime if and only if n = 0,2,4,5,7,8,11,13,16,17,19,31,37,41, 47,53,61,71,79,113,313,353. More recently, H. C. Williams has discoveredthat ^2971,1-503, ¿613^617 and ¿863 are also prime. Williams also states that Í4723and F5387 are probable primes [21].

For Fn to be prime, n > 5, it is necessary, but not sufficient, that n be prime.Similarly, Ln can be prime only when n is prime or a power of 2. There are severalidentities that can be used for primality proofs if one should find either Fn or Ln ortheir primitive parts to be probable primes. These identities are useful because inproving N prime, the methods of [5] depend upon auxiliary factorizations of N± 1.For the Fibonacci numbers we have [9, p. 95]:

(4.1) Fik+i - 1 = FkLkL2k+i, Fik+3 - 1 = -ffc+iLfc+iL^fc+i

and

(4-2) Fik+l + 1 = F2k + lL2k, F4k+3 + 1 = F2k-rlL2k+2-

For the Lucas numbers we have

(4.3) LAk - 1 = L6k/L2k, L4k + l = (L2k - 1)(¿2* + 1)

and

¿4fc+l - 1 = o-FfcLfci^k+i, ¿4fc+3 — 1 = ¿2fc-|-l¿2Jfc+2,(4.4)

¿4/t+i + 1 = L2fcL2fc+i, ¿4t+3 + 1 = bLk+iFk+iF2k+i-

For the Lucas Aurifeuillians we have

A5fc - 1 = 5Fk(Fk - 1), B5fc-l = 5ifc(Ffc + l),A5k + 1 = (Lk-i - l)(Lfc+i - 1), B5k + 1 = (Lk-i + l)(Lfc+i + 1).

The use of these formulas is apparent. They break the factorizations of Fn ± 1 andLn ± 1 into factorizations of smaller Fn's and Ln's and thus facilitate the primalitytest. There are a number of additional formulas of a similar kind for F*±l andLn±l.

All factors and cofactors in Tables 2 and 3 with fewer than 85 digits, and notlabelled as Cxx, have been proved prime by Silverman using the methods presentedin [5, Section 3] and [20]. These methods depend upon auxiliary factorizations ofp — 1, p+1, p2 + l, p2+p-r-l, and p2 - p + 1. If these cyclotomic polynomialshave enough small prime factors, then the methods produce very fast proofs ofprimality along with a compact certificate which can later be used to verify theproof. Andrew Odlyzko has proved all of the remaining probable prime cofactorsto be prime using an implementation of the Cohen-Lenstra algorithm [6].



5. History of Tables. Brillhart found many small factors (up to 10 digits)by a direct search program, using Theorems 2 and 3 to restrict the search rangefor trial division [1], [2]. He later programmed a difference of squares method withmodular exclusion to factor Fi6g, L13í, Lí33, Li34, ¿15s,¿173, and L237.

In 1968 Brillhart used D. H. Lehmer's delay line sieve DLS 127 at U. C. Berkeley[10] to factor ir2gg,Li66,¿2i4,¿252, and L258, again using a difference of squareswith modular exclusion. The most remarkable of these factorizations,

F2*55 = 20778644396941 ■ 20862774425341,was found in just 40 seconds. Although these two factors are very close, there is noknown formula which can account for this factorization.

Between 1970 and 1973, Brillhart and Morrison found a large number of completefactorizations using the continued fraction method, CFRAC, on an IBM 360/91 atUCLA [9], [14].

Starting in 1974, J. L. Selfridge and Marvin C. Wunderlich used an improvedversion of the UCLA program on an IBM 360/65 at NIU in Dekalb, Illinois to factormany 37-41 digit cofactors. They also implemented the first stage of Pollard's just-discovered p — 1 method, and found many new factors. Earl Ecklund and Brillhartprogrammed and used the first stage of the p + 1 method as well [5, p. xlii].

H. C. Williams [19] applied the p ± 1 methods to 174 composite Fibonacci andLucas cofactors which had at most 80 digits.

Thorkil Naur ran the p - 1 and Pollard Rho methods on Fn for odd n, 1 < n <399, and on L„ for 0 < n < 500. When a factor was at most 53 digits, he completedit via CFRAC. His book [15] and paper [16] list several new factorizations whichare included herein.

Montgomery, between 1983 and 1986, applied the methods of [13] to all compositetable entries, using idle time on a VAX/780, two VAX/750's and a CDC 7600. Hefound about 200 previously unknown factors of 11 to 36 digits. Over half of thesewere found by ECM. He used 10 elliptic curves with limits of 104 and 6-105, anotherten curves with limits of 1.6 • 104 and 106, and a third set of ten curves with limitsof 3.2 • 104 and 2 • 106. Often he used four, five or more sets, but the work isuneven (many more curves were used on the Lucas numbers than on the Fibonaccinumbers). Montgomery [13, Section 6] also ran p + 1 with an initial value (seed) of15/8 mod N using limits of 3 • 105 and 107, and again with a seed of 23/11 mod Nusing limits of 2 • 106 and 108. If P = 15/8 mod N, then P2 - 4 = -31/64 mod Nwill be a quadratic residue precisely when -31 is a quadratic residue, so this willfind a factor of p if p - (-^M is highly composite; this includes cases where 31divides whichever of p ± 1 is highly composite. The seed of 23/11 mod N catchescases where p - f | J is highly composite. By Theorems 2 and 3, if p | F* (n odd)

or p I Ln, then p - (|) is divisible by 2n, so the latter case occurs frequently.However, these runs did miss some primes p for which p + 1 is highly composite,such as the factor

2170208701449020077201 = 2 • 7 ■ 12583 ■ 55807 • 424267 • 520309 - 1of Fjgs (found by MP-QS; —31 is a nonresidue, but the limits were not high enoughon that run).



Davis and Holdridge [7], in 1984, completed the factorizations of four cofactors(Í277, ¿362, ¿370> and ¿471) of 57 to 58 digits, using QS on a CRAY IS.

Silverman, between 1983 and 1986, ran p - 1 with limits of 3 • 106 and 5 • 107 onthe entire Lucas table and on the Fibonacci table to F499. He also ran p — 1 withlimits of 2 • 105 and 3 • 106 on the Fibonacci table from Fgni to Fggg. This workwas accomplished on a Micro-VAX/1 and found about 80 new factors. Some runswith ECM on the Lucas table using the same machine revealed no new factors.Silverman also completed the factorizations of all cofactors below 73 digits, andseveral larger ones, using either CFRAC or MP-QS [17], [18] on a combination ofVAX/780's and SUN-3/75's. The larger factorizations were accomplished using aparallel implementation of MP-QS on a network of SUN's.

6. Accuracy and Completeness of Tables. Montgomery and Silvermanindependently verified each entry in the main tables. They checked that

• Each listed factor divides the number and is a prime or probable prime.• The proper list of algebraic (including intrinsic) factors appears• The primitive prime factors appear in ascending order.• If no cofactor is given, the list of factors is complete.• If a cofactor is labelled as Cxx, then it is indeed composite and has xx digits.• If a cofactor is labelled as Pxx, then it is a prime or probable prime and has

xx digits.• No odd primitive prime factor of Fn or L„ was found to divide twice, further

strengthening the conjecture that no such prime exists.Earlier versions of these tables were checked on computers by Michael Morrisonand Tim Korb.

As of August 1987 there remain 140 composite Fibonacci cofactors and 10 com-posite Lucas cofactors in the tables. During 1986 Silverman and Montgomery foundnumerous factors greater than 20 digits, but none smaller. Based upon numerousruns with ECM, the authors are confident that there are at most 3 undetectedfactors less than 20 digits.

7. Discussion of Methods. It is still an open question what the best methodis to attack a large arbitrary composite number. The authors' experience suggeststhat the following procedure is perhaps the most reasonable.

As long as the remaining cofactor TV is not a probable prime, do the followingin order:

(1) Trial division up to some small limit, perhaps (ln A)2.(2) ECM is generally more effective than p ± 1, but p ± 1 is so much faster

that trying it first is worthwhile. A good first set of starting limits is about104 and 105. This should perhaps take a couple of minutes on a typicalmainframe for (say) an 80-digit number.

(3) ECM should now be tried, using about 5 curves and limits of 104 and 5 105.(4) If the remaining cofactor is sufficiently small (say up to 60 digits), it should

be finished with MP-QS. If the number is larger than this, it is worthwhiledevoting more ECM trials with higher limits to it.



(5) If ECM fails and the number is less than about 70 digits, then MP-QSshould now be applied. Seventy digits will take about a day on a typicalmodern mainframe. One can of course attempt larger numbers with asupercomputer or special hardware. The largest number ever factored withMP-QS, as of December 1986, was an 87-digit cofactor of 5128 + 1 using aparallel implementation on a SUN network. That factorization took 3950total CPU hours, divided among 10 SUN-3's over a period of about 5 weeks.

(6) Finally, if the cofactor is still too large, one can keep trying ECM withhigher limits or set the number aside.

TABLE 1

Prime Factors With More Than 25 Digits

N Factor Discoverer Method Machine¿386¿563

¿431¿425

¿507¿406

¿371

¿422¿467

¿320

¿837¿445

¿471¿277¿517

¿741

¿597¿503

¿869¿479

¿559¿317¿461

¿633

¿326¿971¿412¿344¿482¿377¿489

¿663¿681¿430¿383¿427¿464

1024502971279512003440504312158771296959377863294133137804955311272103560184211418795434530356438839000117340889195212892399797173236706989118808657589803873566879698948480066612280936302689192832119042589867473810531747821913958970316237955583180309986727296113629977270254443767966033315652528928254841408179908147833025812336055419986916950547100574069152485329362164146612434960769701623809766965207227105127169843611312262448164949728997664039492430014747700999423017017501589043082120466508853546991316372649304949588683920725489260938370570172472695312215215035463301653338050423852190957907365333787128886141126177192347474285460831200493920089573005680996120855900783871963619802607259514583330235693729109041433538316846356114516762314039810997233210293799139486415373430329122468821883671012169922040724372371994215431788839955010483350408487052485570744297542202788462733966380018208818089131653446329084721859009751356451315174165446397191756452643802471611531850844381077461461960364348676924949586896499848287125235667356281227693725298545340302283668318476481

MontgomeryMontgomerySilvermanSilvermanSilvermanSilvermanSilvermanSilvermanMontgomeryNaurSilvermanSilvermanDavisDavisSilvermanSilvermanSilvermanSilvermanMontgomerySilvermanSilvermanSilvermanSilvermanSilvermanSilvermanMontgomeryMontgomerySilvermanMontgomerySilvermanSilvermanSilvermanSilvermanSilvermanSilvermanSilvermanMontgomery

ECMECMp-1MP-QSMP-QSMP-QSMP-QSMP-QSECMCFRACMP-QSMP-QSQSQSMP-QSMP-QSMP-QSMP-QSECMMP-QSMP-QSMP-QSMP-QSMP-QSMP-QSP - (5/p)ECMMP-QSECMMP-QSMP-QSMP-QSMP-QSMP-QSMP-QSMP-QSECM

CDC 7600CDC 7600UVAX/1VAX/780VAX/8600VAX/780VAX/780SUN-3/75CDC 7600MathildaSUN-3/75VAX/780CRAY ISCRAY ISSUN-3/75SUN-3/75SUN-3/75SUN-3/75CDC 7600VAX/780SUN-3/75VAX/780SUN-3/75SUN-3/75SUN-3/75CDC 7600CDC 7600VAX/780CDC 7600SUN-3/75SUN-3/75SUN-3/75SUN-3/75SUN-3/75SUN-3/75SUN-3/75CDC 7600



The present practical limit of technology seems to be about 16 digits for primefactors found by Pollard Rho, 18 digits for Brent's variation of Pollard Rho, and 25digits for ECM. The p± 1 methods occasionally have huge successes where a factorover 25 digits is found; for example, these methods could have found the 29-digitfactor of L479 with a little more effort. However, factors of 18 to 20 digits are moretypical. The CFRAC method has been demonstrated for products up to 1064, QSfor products up to 1071, and MP-QS for products up to 1087. This comparison isnot quite fair, however, because the CFRAC and QS results were achieved eitheron a supercomputer or on special purpose hardware, while the MP-QS results wereachieved on a network of SUN's [17], [18].

Table 1 lists all of the known nonlargest primitive prime factors of Fn or Lnhaving more than 25 digits. The cofactor of each of these, when it is composite, isassumed to have at least one prime factor exceeding the factor listed. Each entryincludes the discoverer, the method of discovery, and the machine used. In the"machine" column the notation "UVAX/1" is an abbreviation for Micro-VAX/1.

Acknowledgments. The authors wish to thank their institutions for providingthe extensive computer time required for this work. Silverman extends specialthanks to Tom Caron for writing the networking software that made the parallelimplementation of MP-QS possible. Montgomery extends special thanks to MikeMitchell and Richard Milton for the many hours they spent running his CDC 7600jobs. All of us extend special thanks to Andrew Odlyzko for providing many of ourprimality tests and to Dave Cantor, Tim Korb, D.H. and Emma Lehmer, MichaelMorrison, J. L. Selfridge, Peter Weinberger, Hugh C. Williams, and Marvin C.Wunderlich for their assistance over the years in helping to create these tables.

Department of MathematicsUniversity of ArizonaTucson, Arizona 85721

UNISYS2525 Colorado AvenueSanta Monica, California 90406

MITRE CorporationBurlington RoadBedford, Massachusetts 01730

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 factoriza-

tions of 2m ± 1," Math. Comp., v. 29, 1975, pp. 620-647.4. J. BRILLHART, "UMT of F„ for n < 3500 and L„ for n < 1750," Math. Comp., submitted

to unpublished tables.5. J. BRILLHART, D. H. LEHMER, J. L. SELFRIDGE, B. TUCKERMAN & S. S. WAGSTAFF,

Jr., Factorizations of bn ± 1, b — 2,3,5,6,7,10,11,12 Up to High Powers, Contemp. Math., vol.22, Amer. Math. Soc, Providence, R.I., 1983.

6. H. COHEN & H. W. LENSTRA, Jr., "Primality testing and Jacobi sums," Math. Comp., v.42, 1984, pp. 297-330.

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.

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. 519-521.13. PETER L. MONTGOMERY, "Speeding the Pollard and elliptic curve methods of factoriza-

tion," Math. Comp., v. 48, 1987, pp. 243-264.14. M. A. Morrison & J. Brillhart, "A method of factoring and the factorization of Ft,"

Math. Comp., v. 29, 1975, pp. 183-205.15. 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. 687-695.17. ROBERT D. SILVERMAN, "The multiple polynomial quadratic sieve," Math. Comp., v. 48,

1987, pp. 329-339.18. ROBERT D. SILVERMAN, "Parallel implementation of the quadratic sieve," The Journal of

Supercomputing, v. 1, 1987, no. 3.19. H. C. WILLIAMS, " A p + 1 method of factoring," Math. Comp., v. 39, 1982, pp. 225-234.20. H. C. WILLIAMS ic J. S. JUDD, "Some algorithms for prime testing using generalized

Lehmer functions," Math. Comp., v. 30, 1976, pp. 867-886.21. H. C. WILLIAMS, private communication.


MATHEMATICS OF COMPUTATIONVOLUME 50, NUMBER 181JANUARY 1988, PAGES S1-S15

Supplement toTables of Fibonacci and Lucas Factorizations

By John Brillhart, Peter L. Montgomery and Robert D. Silverman

TABLE 2 FIBONACCI FACTORIZATIONS 2 < n< 1000, n odd

n Prime Factors

3 2557 139 (3) 17

11 8913 23315 (3,5) 6117 159719 37.11321 (3,7) 42123 2865725 (5) 5*.300127 (3,9) 53.10929 51422931 557.241733 (3,11) 1980135 (5,7) 14196137 73.149.222139 (3,13) 13572141 2789.5936943 43349443745 (3,5,9,15) 10944147 297121507349 (7) 97.616870951 (3,17) 637602153 953.5594574155 (5,11) 661.47454157 (3,19) 797.5483359 353.271026069761 4513.55500349763 (3,7,9,21) 3523968165 (5,13) 1473620616167 269.116849.142991369 (3,23) 137.829.1807771 6673.4616537107373 9375829.8602071775 (3,5,15,25) 23068650177 (7,11) 988681.4832521

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tables of fibonacci and lucas factorizations · 2018. 11. 16. · mathematics of computation volume...

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