# Pell and Lucas primes

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1 Notes o Number Theory ad Discrete Mathematics ISSN Vol. 2, 205, No. 3, 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 s: 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 Table : Some Pell umbers 64

2 2 Factor fuctios Ulike the Lucas umbers, the Pell umbers [6] satisfy that is, 2 2 F = F+ + F+ ; (2.) ( P ) ( P ). P (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 p p c 3 3; 3 = 2p p p c ; 37 = 8p + ; 8297 = 488p p c ; 229 = 0p + ; = 42730p p c ; 6 = 2p ; = p p 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

3 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 = P R, (2.6) ad P P 4S, 2 (2.7) = 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, = 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 p c p c p c p p Table 3: Uit sigs i Equatio (3.2) 66