Notes o Number Theory ad Discrete Mathematics ISSN 30 532 Vol. 2, 205, No. 3, 64 69 Pell ad Lucas primes J. V. Leyedekkers ad A. G. Shao 2 Faculty of Sciece, The Uiversity of Sydey NSW 2006, Australia 2 Faculty of Egieerig & IT, Uiversity of Techology, Sydey NSW 2007, Australia e-mails: t.shao@warrae.usw.edu.au, Athoy.Shao@uts.edu.au Abstract: The structures of Pell ad Lucas umbers, P p ad L p with prime subscripts are compared i relatio to the fuctio (Kp ± ) ad for factors of the form (kp ± ). It is foud that digit sums give some guides to primality. Keywords: Pell umbers, Lucas umbers, Primality, Digit sums. AMS Classificatio: B39, B50. Itroductio We have recetly cosidered some methods for checkig the primality of Fiboacci umbers [5 9]. For some of the earlier history see []. Sice the Pell ad Lucas umbers are structurally similar to the Fiboacci umbers it makes sese to attempt to apply the same methods to these umbers. The Pell umbers are geerated from the secod order liear homogeeous recurrece relatio P 2, = P + P 2 0. (.) ad with suitable iitial terms we obtai (Table ). P P P P 0 0 574 2 3863965 3 25977522849 2 3860 22 93222358 32 62703566048 2 2 3 3346 23 22505868 33 53744654945 3 5 4 80782 24 543339720 34 3654502875938 4 2 5 95025 25 37382 35 8822755040682 5 29 6 470832 26 36685962 36 2300003689580 6 70 7 36689 27 7645370045 37 542275778598 7 69 8 274420 28 8457556052 38 24455926542 8 408 9 662509 29 4456048249 39 29973796309065 9 985 20 5994428 30 07578520350 40 723573879672 0 2378 4 746860020068409 Table : Some Pell umbers 64
2 Factor fuctios Ulike the Lucas umbers, the Pell umbers [6] satisfy that is, 2 2 F = F+ + F+ ; (2.) 2 2 2 2 ( P ) ( P ). P (2.2) = 2 + + 2 It might be expected the that the Fiboacci factor fuctio ( F ) = kp ± f p (2.3) would also exted to composite prime-subscripted Pell umbers which it does (Table 2). Digit p P p sum Type Factors 3 5 5 p --- 5 29 2 p --- 7 69 7 c 3 3; 3 = 2p 574 8 p --- 3 3346 8 p --- 7 36689 7 c 37 8297; 37 = 8p + ; 8297 = 488p + 9 662509 2 p --- 23 22505868 c 229 982789; 229 = 0p + ; 982789 = 42730p 29 4456048249 2 p --- 3 25977522849 7 c 6 4257664309; 6 = 2p ; 4257664309 = 3734400p 37 542275778598 8 p --- 4 746860020068409 7 p --- Table 2: Factors of some prime-subscripted Pell umbers [Factorisatios checked with Mathematica ad WolframAlpha] The prime-subscripted Pell umbers i Table 2 also satisfy the recurrece relatio P2 + = 6P2 P2 3, >. (2.4) with iitial coditios ad 5. They are also worthy of ote i this cotext because they are related to the Pell idetities of Horadam, [3] ad the Pythagorea triads of Forget ad Larki, [2] through related Pell-type sequeces {R } ad {S }. For example, 2 2 2P = R, (2.5) 2 65
i which {R } also satisfies (2.5) but with iitial coditios R = ad R 2 = 7. From (2.5) we ca also obtai P 2+ 5 = P2 + + 4R, (2.6) ad P P 4S, 2 (2.7) 2 + 4 = 2 + + where {S } also satisfies (2.7) but with iitial coditios S = ad S 2 = 3, so that 2S R R. (2.8) = It is also worthy of ote that umbers which satisfy (2.4) with iitial coditios (, 6) or (3, 7) are called balacig umbers: a iteger is called a balacig umber (or a Lucasbalacig umber) with balacer r if it is the solutio of the Diophatie equatio [2] j= j = + r j= + For example, + 2 + 3 + 4 + 5 = 5 = 7 + 8, so that 6 is a balacig umber with balacer 2. j. 3 K fuctio The Fiboacci umber, F p, may be expressed as F p = Kp ± (3.) i which K is a fuctio of the sum of p cosecutive Fiboacci umbers [9]. The digit sum of K yields a primality check [8]. Correspodigly (Table 3) P p = Kp ± (3.2) p p c K Kp + Kp 3 p 2 5 p 6 7 c 24 p 522 3 p 2574 7 c 66864 9 p 348690 23 c 978560 29 p 536568350 3 c 8377984608 37 p 389804264486 4 p 4260634952888 Table 3: Uit sigs i Equatio (3.2) 66
The sig for i Equatio (3.2) is geerally egative for primes ad positive for composites for the rage cosidered here. The values for the digit sum of K for Pell umbers exhibit a commo sum whe the right-ed-digit (RED) (or value (modulo 0)) p* = 3, so K is ot as cosistet as it is for F p (Table 4). p* p c p* p c 3,9 6 7,8 7 3 2,9 9 3 5,8 7 9 3,6 7 8 7 9 3,6 --- 9 2 --- Table 4: Sum of digit of K (from Equatio (3.2)) Table 5: Sum of digits of P p (from Table ) Furthermore, the digit sums of P p show clearer distictios tha K betwee primes ad composites (Table 5). 4 Lucas primes The Lucas umbers are similar i structure to the Fiboacci umbers i may ways, but particularly i the cotext of this paper i that they cosist of the odd-odd-eve form, whereas the Pell umbers follow a eve-odd patter. However, the Lucas primes are ot restricted to Lucas umbers with a prime subscript; for example, L 8 = 47 ad L 6 = 2207, both of which are prime umbers. O the other had, the umber L = K ± (4.) is essetially the same as for F p ad P p. For istace, L 8 = 6 = 47; L6 = 38 = 2207; L = 8 + = 99. Moreover, for composite L p the factors have the patter (kp ± ) as for the Fiboacci ad Pell sequeces, [8]. For example, L 23 = 39 46 = (6 p + )(20 p + ). Why the do the elemets of the Lucas sequece fail to follow Equatios (2.2) or (2.3)? Noe of the Lucas umbers have REDs equal to 0 or 5; that is, oe is i the class 05 Z 5 [0], which meas that costraits occur. * If L {3, 7}, the the values of ½( + ) ad ( ) caot be itegers as is always eve, so that L with these REDs caot satisfy Equatios (2.2) or (2.3); * If L {, 9}, the the values of ½( + ) ad ½( ) yield d 2, e 2 couples of (, 4) or (9, 6) ad L REDs are 5 so that Equatios (2.2) ad (2.3) caot be satisfied. Primes have oe set of (d,e) with o commo factors [4] ad for Pell ad Fiboacci umbers this set is give by Equatios (2.2) ad (2.3). Composites have as may (d, e) couples 67
as there are factors. These restrictios permit a further test for primality, which is discussed i * detail elsewhere [4], ad this structure explais why some primes, L = 3, 7 have eve. 5 Fial commets It has bee foud here that the Lucas umbers show differeces from the Pell ad Fiboacci umbers i the formatio of primes. The results are related to those of Mc Daiel [], which build o the results of Robbis [4, 5] ad are related to Ribeboim s, [3]. The results here provide further isights ito the structure of the prime system. The fuctios (pk ± ) ad (pk ± ) are idepedet of the triples formed from the Fiboacci, Pell ad Lucas sequeces. The overall prime structure which applies to the subscripts of the three sets of triples must relate to the patter of primes versus composites formed. I other words, here are clues to the master prime structure. Further aalysis alog the lies we have bee followig [4 9] might well lik up with other prime-structure research. Refereces [] Duber, H., & Keller, W. (999) New Fiboacci ad Lucas Primes. Mathematics of Computatio. 68, 47 427. [2] Forget, T. W., & Larki, T. A. (968). Pythagorea Triads of the form x, x +, z described by Recurrece Sequeces. The Fiboacci Quarterly 6 (3): 94 04. [3] Horadam, A.F. (97) Pell Idetities. The Fiboacci Quarterly. 9 (3): 245 252, 263. [4] Leyedekkers, J. V., & Shao, A. G. (998) Fiboacci Numbers withi Modular Rigs. Notes o Number Theory ad Discrete Mathematics. 4 (4): 65 74. [5] Leyedekkers, J. V., & Shao, A. G. (203) The Structure of the Fiboacci Numbers i the Modular Rig Z 5. Notes o Number Theory ad Discrete Mathematics. 9(), 66 72. [6] Leyedekkers, J. V., & Shao, A. G. (203) Fiboacci ad Lucas Primes. Notes o Number Theory ad Discrete Mathematics. 9(2): 49 59. [7] Leyedekkers, J. V., & Shao, A. G. (204) Fiboacci Primes. Notes o Number Theory ad Discrete Mathematics. 20(2), 6 9. [8] Leyedekkers, J. V., & Shao, A. G. (204) Fiboacci Numbers with Prime Subscripts: Digital Sums for Primes versus Composites. Notes o Number Theory ad Discrete Mathematics. 20(3), 45 49. [9] Leyedekkers, J. V., & Shao, A. G. (204) Fiboacci Number Sums as Prime Idicators. Notes o Number Theory ad Discrete Mathematics, 20(4), 47 52. [0] Leyedekkers, J. V., Shao, A. G. & Rybak, J.M. (2007) Patter Recogitio: Modular Rigs ad Iteger Structure. North Sydey: Raffles KvB Moograph No.9. 68
[] McDaiel, W. (2002) O Fiboacci ad Pell Numbers of the form kx 2 : Almost every term has a 4r + prime factor. The Fiboacci Quarterly. 40(), 4 42. [2] Ray, P. K. (203) New Idetities for the Commo Factors of Balacig ad Lucas- Balacig umbers. It. J. of Pure ad Applied Mathematics. 85(3), 487 494. [3] Ribeboim, P. (999) Pell Numbers, Squares ad Cubes. Publicatioes Mathematicae- Debrece. 54( 2), 3 52. [4] Robbis, N. (983) O Fiboacci Numbers of the Form PX 2 where P is Prime. The Fiboacci Quarterly. 2(3), 266 27. [5] Robbis, N. (984) O Pell Numbers of the Form PX 2 where P is Prime. The Fiboacci Quarterly. 22(4), 340 348. [6] Watkis, J J. (204) Number Theory: A Historical Approach. Priceto ad Oxford: Priceto Uiversity Press, pp. 27 272. 69