Tables of Fibonacci and Lucas Factorizations

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

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

Abstract. We list the known prime factors of the Fibonacci numbers Fn for n < 999 and Lucas numbers Ln for n < 500. We discuss the various methods used to obtain these 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 in two tables the known prime factors of the Fibonacci numbers Fn, 3 < n < 999, n odd, and the Lucas numbers Ln, 2 < n < 500. The sequences Fn and Ln are defined 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 these numbers. We also have the formulas an - 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 the multiplicative structure of Fn and Ln. It includes both theoretical and numerical results.

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 defined additively, 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.

251

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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 part ofFn. Formula (2.4) reduces factoring Fn to factoring the FJ's. Formula (2.5) shows that 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 is also called algebraic if it is a product of prime algebraic factors. Any prime divisor of F¿ (resp. L'n) is necessarily algebraic, but under certain circumstances a prime divisor 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 exactly when n = prm, r > 1, where p is a primitive factor of Fm (resp. Lm). In this case p always divides F* (resp. L* ) to just the first power. Example. The factorization of F105, given by (2.4), is

^105-j? _ [I1 f I'djr>* -•i,l-f3^5-f7^15^21-I'35-f105-_ jp* jp* ¡p* jp* jp* jp* jp* jp* 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 algebraic factors 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 to the numbers inside the parentheses are: 32 55 7 13 15 (3,5)61 21 (3.7)421 35 (5,7) 141961 The factorization of Fins is then obtained by collecting the primitive prime factors from their respective lines. These follow the parentheses (if any) on the seven 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 derived directly from that of F2n. Let n = 2sm, where m is odd. Then (2.6) Ln = n¿2M, «>1, d\m

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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 discovered by Aurifeuille, we also have for odd n

i^n = a +p = a4n _ a3nßn + a2n/32n _ anß3n + /?4n Ln 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, where A5„ = 5Fn2- 5Fn + 1, B5n = 5F2 + 5Fn + 1. This decomposition means that these ¿5n's have two different algebraic factorizations. For example, from (2.6) and (2.9)

¿ios = il Ld = LXL3L5L7LX5L2XL35L105 d|105 and ¿105 = ¿2lAi05.Bi05- Primitive parts An and ß* can also be defined for An and Bn. Let n > 1 be odd and set n = 5"m,s > 0,5 f ra. Let £d — \ (l + (§)), where (g) is the Legendre symbol. Let

(2-10) ÎÎ Bin=\[[(AsnldY-*<(Bsn/dyr{d)- d\m

(Here A%n and B$n are rational integers such that (A%n,Bln) = 1 and L%n = A%nBln.) Then Mn=X[(Aln/dY<(Bln/dy-*<,

d\m Thus, in the above example we have

Al05 = A5Î?Î5.B35AÎ05, B105 = Bg AJgAggBJng.

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Since j4g = A\5 = 1, these are omitted in Table 3, while Bg" is written as Lg and BU as L*X5. Those Lucas numbers which do not have an Aurifeuillian factorization appear in the tables in the same format as the Fibonacci factorizations. However, the Aurifeuillian factorizations appear in an expanded format. For example, the above factorization appears as: 105 (3,7,21) AB A (15,35B) 21211 B (5,35A) 767131. The list of numbers immediately after the index 105 indicate that L105 has the algebraic factors L\,L7, and L\x. Furthermore, A\QX>has algebraic factors L*X5and B35, while Biog has algebraic factors L*band A35. In computing A* and B*, the following 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 prime factor appears explicitly more than once in the tables (except intrinsic factors and the repeated factor 2 of L3). Where space permits, we list the known factors in their entirety on a single line. We list all prime factors of 25 digits or less, carrying over to a second line, without breaking the factor, when necessary. All other factors are listed as either Pxx or Cxx, indicating respectively a prime or a composite cofactor of xx digits. When a factorization is incomplete, we leave space on the line for new factors to be inserted by hand. 3. Factorization Methods. A variety of methods have been used to effect the factorizations given herein. These include the Pollard p - 1 and Brent-Pollard Rho methods [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's Elliptic Curve Method (ECM) [11], [13]. Of course, many of the smaller prime factors are quite old, and were originally found by trial division or the difference of squares method. Some of the methods utilize the form of the prime divisors given by the following theorems [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.

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THEOREM 3. Let n be positive and let p be an odd, primitive prime divisor of Ln. 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 primality tests on the Fibonacci and Lucas numbers: Fn, 3 < n < 1000, is prime if and only 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. Williamshas discovered that ^2971,1-503, ¿613^617 and ¿863 are also prime. Williams also states that Í4723 and 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 several identities that can be used for primality proofs if one should find either Fn or Ln or their primitive parts to be probable primes. These identities are useful because in proving 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 and Ln ± 1 into factorizations of smaller Fn's and Ln's and thus facilitate the primality test. There are a number of additional formulas of a similar kind for F*±l and Ln±l. All factors and cofactors in Tables 2 and 3 with fewer than 85 digits, and not labelled as Cxx, have been proved prime by Silverman using the methods presented in [5, Section 3] and [20]. These methods depend upon auxiliary factorizations of p — 1, p+1, p2 + l, p2+p-r-l, and p2 - p + 1. If these cyclotomic polynomials have enough small prime factors, then the methods produce very fast proofs of primality along with a compact certificate which can later be used to verify the proof. Andrew Odlyzko has proved all of the remaining probable prime cofactors to be prime using an implementation of the Cohen-Lenstra algorithm [6].

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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 range for trial division [1], [2]. He later programmed a difference of squares method with modular 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 squares with 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 no known formula which can account for this factorization. Between 1970 and 1973, Brillhart and Morrison found a large number of complete factorizations using the continued fraction method, CFRAC, on an IBM 360/91 at UCLA [9], [14]. Starting in 1974, J. L. Selfridge and Marvin C. Wunderlich used an improved version of the UCLA program on an IBM 360/65 at NIU in Dekalb, Illinois to factor many 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 Brillhart programmed 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 and Lucas 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 completed it via CFRAC. His book [15] and paper [16] list several new factorizations which are included herein. Montgomery, between 1983 and 1986, applied the methods of [13] to all composite table entries, using idle time on a VAX/780, two VAX/750's and a CDC 7600. He found about 200 previously unknown factors of 11 to 36 digits. Over half of these were found by ECM. He used 10 elliptic curves with limits of 104 and 6-105, another ten curves with limits of 1.6 • 104 and 106, and a third set of ten curves with limits of 3.2 • 104 and 2 • 106. Often he used four, five or more sets, but the work is uneven (many more curves were used on the Lucas numbers than on the Fibonacci numbers). Montgomery [13, Section 6] also ran p + 1 with an initial value (seed) of 15/8 mod N using limits of 3 • 105 and 107, and again with a seed of 23/11 mod N using limits of 2 • 106 and 108. If P = 15/8 mod N, then P2 - 4 = -31/64 mod N will be a quadratic residue precisely when -31 is a quadratic residue, so this will find a factor of p if p - (-^M is highly composite; this includes cases where 31 divides whichever of p ± 1 is highly composite. The seed of 23/11 mod N catches cases 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- 1 of Fjgs (found by MP-QS; —31 is a nonresidue, but the limits were not high enough on that run).

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

6. Accuracy and Completeness of Tables. Montgomery and Silverman independently 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 Morrison and Tim Korb. As of August 1987 there remain 140 composite Fibonacci cofactors and 10 composite Lucas cofactors in the tables. During 1986 Silverman and Montgomery found numerous factors greater than 20 digits, but none smaller. Based upon numerous runs with ECM, the authors are confident that there are at most 3 undetected factors less than 20 digits.

7. Discussion of Methods. It is still an open question what the best method is to attack a large arbitrary composite number. The authors' experience suggests that the following procedure is perhaps the most reasonable. As long as the remaining cofactor TV is not a probable prime, do the following in 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 about 104 and 105. This should perhaps take a couple of minutes on a typical mainframe 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 worthwhile devoting more ECM trials with higher limits to it.

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(5) If ECM fails and the number is less than about 70 digits, then MP-QS should now be applied. Seventy digits will take about a day on a typical modern mainframe. One can of course attempt larger numbers with a supercomputer or special hardware. The largest number ever factored with MP-QS, as of December 1986, was an 87-digit cofactor of 5128 + 1 using a parallel implementation on a SUN network. That factorization took 3950 total 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 with higher limits or set the number aside.

TABLE 1 Prime Factors With More Than 25 Digits

N Factor Discoverer Method Machine ¿386 10245029712795120034405043 Montgomery ECM CDC 7600 ¿563 12158771296959377863294133 Montgomery ECM CDC 7600 ¿431 13780495531127210356018421 Silverman p-1 UVAX/1 ¿425 14187954345303564388390001 Silverman MP-QS VAX/780 ¿507 17340889195212892399797173 Silverman MP-QS VAX/8600 ¿406 23670698911880865758980387 Silverman MP-QS VAX/780 ¿371 35668796989484800666122809 Silverman MP-QS VAX/780 ¿422 36302689192832119042589867 Silverman MP-QS SUN-3/75 ¿467 47381053174782191395897031 Montgomery ECM CDC 7600 ¿320 62379555831803099867272961 Naur CFRAC Mathilda ¿837 136299772702544437679660333 Silverman MP-QS SUN-3/75 ¿445 156525289282548414081799081 Silverman MP-QS VAX/780 ¿471 478330258123360554199869169 Davis QS CRAY IS ¿277 505471005740691524853293621 Davis QS CRAY IS ¿517 641466124349607697016238097 Silverman MP-QS SUN-3/75 ¿741 669652072271051271698436113 Silverman MP-QS SUN-3/75 ¿597 1226244816494972899766403949 Silverman MP-QS SUN-3/75 ¿503 2430014747700999423017017501 Silverman MP-QS SUN-3/75 ¿869 5890430821204665088535469913 Montgomery ECM CDC 7600 ¿479 16372649304949588683920725489 Silverman MP-QS VAX/780 ¿559 26093837057017247269531221521 Silverman MP-QS SUN-3/75 ¿317 50354633016533380504238521909 Silverman MP-QS VAX/780 ¿461 57907365333787128886141126177 Silverman MP-QS SUN-3/75 ¿633 192347474285460831200493920089 Silverman MP-QS SUN-3/75 ¿326 573005680996120855900783871963 Silverman MP-QS SUN-3/75 ¿971 619802607259514583330235693729 Montgomery P - (5/p) CDC 7600 ¿412 1090414335383168463561145167623 Montgomery ECM CDC 7600 ¿344 1403981099723321029379913948641 Silverman MP-QS VAX/780 ¿482 5373430329122468821883671012169 Montgomery ECM CDC 7600 ¿377 9220407243723719942154317888399 Silverman MP-QS SUN-3/75 ¿489 55010483350408487052485570744297 Silverman MP-QS SUN-3/75 ¿663 542202788462733966380018208818089 Silverman MP-QS SUN-3/75 ¿681 1316534463290847218590097513564513 Silverman MP-QS SUN-3/75 ¿430 1517416544639719175645264380247161 Silverman MP-QS SUN-3/75 ¿383 15318508443810774614619603643486769 Silverman MP-QS SUN-3/75 ¿427 24949586896499848287125235667356281 Silverman MP-QS SUN-3/75 ¿464 227693725298545340302283668318476481 Montgomery ECM CDC 7600

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The present practical limit of technology seems to be about 16 digits for prime factors found by Pollard Rho, 18 digits for Brent's variation of Pollard Rho, and 25 digits for ECM. The p± 1 methods occasionally have huge successes where a factor over 25 digits is found; for example, these methods could have found the 29-digit factor of L479 with a little more effort. However, factors of 18 to 20 digits are more typical. The CFRAC method has been demonstrated for products up to 1064, QS for products up to 1071, and MP-QS for products up to 1087. This comparison is not quite fair, however, because the CFRAC and QS results were achieved either on a supercomputer or on special purpose hardware, while the MP-QS results were achieved on a network of SUN's [17], [18]. Table 1 lists all of the known nonlargest primitive prime factors of Fn or Ln having more than 25 digits. The cofactor of each of these, when it is composite, is assumed to have at least one prime factor exceeding the factor listed. Each entry includes 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 providing the extensive computer time required for this work. Silverman extends special thanks to Tom Caron for writing the networking software that made the parallel implementation of MP-QS possible. Montgomery extends special thanks to Mike Mitchell and Richard Milton for the many hours they spent running his CDC 7600 jobs. All of us extend special thanks to Andrew Odlyzko for providing many of our primality tests and to Dave Cantor, Tim Korb, D.H. and Emma Lehmer, Michael Morrison, 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 Mathematics University of Arizona Tucson, Arizona 85721 UNISYS 2525 Colorado Avenue Santa Monica, California 90406 MITRE Corporation Burlington Road Bedford, 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 factorizations 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.

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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 factorization," 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.

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Supplement to Tables 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 2 55 7 13 9 (3) 17 11 89 13 233 15 (3,5) 61 17 1597 19 37.113 21 (3,7) 421 23 28657 25 (5) 5*.3001 27 (3,9) 53.109 29 514229 31 557.2417 33 (3,11) 19801 35 (5,7) 141961 37 73.149.2221 39 (3,13) 135721 41 2789.59369 43 433494437 45 (3,5,9,15) 109441 47 2971215073 49 (7) 97.6168709 51 (3,17) 6376021 53 953.55945741 55 (5,11) 661.474541 57 (3,19) 797.54833 59 353.2710260697 61 4513.555003497 63 (3,7,9,21) 35239681 65 (5,13) 14736206161 67 269.116849.1429913 69 (3,23) 137.829.18077 71 6673.46165371073 73 9375829.86020717 75 (3,5,15,25) 230686501 77 (7,11) 988681.4832521

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