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K-Fibonacci Sequences Modulo M Available online at www.sciencedirect.com Chaos, Solitons and Fractals 41 (2009) 497–504 www.elsevier.com/locate/chaos k-Fibonacci sequences modulo m Sergio Falcon *,A´ ngel Plaza Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), 35017 Las Palmas de Gran Canaria, Spain Accepted 14 February 2008 Abstract We study here the period-length of the k-Fibonacci sequences taken modulo m. The period of such cyclic sequences is know as Pisano period, and the period-length is denoted by pkðmÞ. It is proved that for every odd number k, 2 2 pkðk þ 4Þ¼4ðk þ 4Þ. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Many references may be givenpffiffi for Fibonacci numbers and related issues [1–9]. As it is well-known Fibonacci num- 1þ 5 bers and Golden Section, / ¼ 2 , appear in modern research in many fields from Architecture, Nature and Art [10– 16] to theoretical physics [17–21]. The paper presented here is focused on determining the length of the period of the sequences obtained by reducing k- Fibonacci sequences taking modulo m. This problem may be viewed in the context of generating random numbers, and it is a generalization of the classical Fibonacci sequence modulo m. k-Fibonacci numbers have been ultimately intro- duced and studied in different contexts [22–24] as follows: Definition 1. For any integer number k P 1, the kth Fibonacci sequence, say fF k;ngn2N is defined recurrently by: F k;0 ¼ 0; F k;1 ¼ 1; and F k;nþ1 ¼ kF k;n þ F k;nÀ1 for n P 1: Note that if k is a real variable x then F k;n ¼ F x;n and they correspond to the Fibonacci polynomials defined by: 8 9 <> 1ifn ¼ 0 => F ðxÞ¼ x if n ¼ 1 : nþ1 :> ;> xF nðxÞþF nÀ1ðxÞ if n > 1 Particular cases of the previous definition are: If k ¼ 1, the classical Fibonacci sequence is obtained: F 0 ¼ 0; F 1 ¼ 1, and F nþ1 ¼ F n þ F nÀ1 for n P 1 : fF ngn2N ¼ f0; 1; 1; 2; 3; 5; 8; ...g: * Corresponding author. Tel.: +34 928 45 88 27; fax: +34 928 45 87 11. E-mail address: [email protected] (S. Falcon). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.02.014 498 S. Falcon, A´ . Plaza / Chaos, Solitons and Fractals 41 (2009) 497–504 If k ¼ 2, the Pell sequence appears: P 0 ¼ 0; P 1 ¼ 1, and P nþ1 ¼ 2P n þ P nÀ1 for n P 1: fP ngn2N ¼f0; 1; 2; 5; 12; 29; 70; ...g: If k ¼ 3, the following sequence appears: F 3;0 ¼ 0; F 3;1 ¼ 1, and F 3;nþ1 ¼ 3F 3;n þ F 3;nÀ1 for n P 1: fF 3;ngn2N ¼f0; 1; 3; 10; 33; 109; ...g: From the definition of the k-Fibonacci numbers, the first of them are presented in Table 1, and from these expres- sions, one may deduce the value of any k-Fibonacci number by simple substitution on the corresponding F k;n. For example, the seventh element of the 4-Fibonacci sequence, fF 4;ngn2N ,isF 4;7 ¼ 46 þ 5 Á 44 þ 6 Á 42 þ 1 ¼ 5473. By doing k ¼ 1; 2; 3; ... the respective k-Fibonacci sequences are obtained: fF 1;ngn2N ¼f0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; ...g; fF 2;ngn2N ¼f0; 1; 2; 5; 12; 29; 70; 169; 408; 985; 2378; ...g; fF 3;ngn2N ¼f0; 1; 3; 10; 33; 109; 360; 1189; 3927; 12970; 42837; ...g; fF 4;ngn2N ¼f0; 1; 4; 17; 72; 305; 1292; 5473; 23184; 98209; 416020; ...g: Some of the properties the k-Fibonacci sequences verify are summarized bellow, (see [22–24] for details of the proofs): pffiffiffiffiffiffiffi rnðrÞÀn kþ k2þ4 [Binet’s formula] F k;n ¼ rþrÀ1 , where r ¼ 2 is the positive root of the characteristic equation associated to the recurrence relation defining k-Fibonacci numbers: r2 À kr À 1 ¼ 0. [First combinatorial formula for the general term] nÀ1 bcX2 1 n nÀ1À2i 2 i F k;n ¼ nÀ1 k ðk þ 4Þ : 2 i¼0 2i þ 1 [Second combinatorial formula for the general term] nÀ1 bcX2 n À 1 À i nÀ1À2i F k;n ¼ k : i¼0 i nþ1Àr [Catalan’s identity] F k;nÀrF k;nþr À F k;n2 ¼ðÀ1Þ F k;r2 n [Simson’s identity] F k;nÀ1F k;nþ1 À F k;n2 ¼ðÀ1Þ n [d’Ocagne’s identity] F k;mFPk;nþ1 À F k;mþ1F k;n ¼ðÀ1Þ F k;mÀn n 1 [Sum of the first n terms] i¼1FPk;i ¼ k ðF k;nþ1 þ F k;n À 1Þ n 1 [Sum of the first n even terms]P i¼1F k;2i ¼ k ðF k;2nþ1 À 1Þ n 1 [Sum of the first n odd terms] i¼0F k;2iþ1 ¼ k F k;2nþ2 x [Generating function] fkðxÞ¼1ÀkxÀx2 2. Pisano periods for the k-Fibonacci sequences In this Section, the sequences obtained by considering the remainders of the k-Fibonacci sequences modulo m, being m an integer, are considered. For any fixed integer m, and varying the value of parameter k ¼ 1; 2; ...a new sequence is obtained by considering the remainders modulo m. The nth Pisano period, written pðnÞ, is the period with which the sequence of k-Fibonacci numbers, modulo n repeats [25]. Table 1 The first k-Fibonacci numbers F k;1 ¼ 1 F k;2 ¼ k 2 F k;3 ¼ k þ 1 3 F k;4 ¼ k þ 2k 4 2 F k;5 ¼ k þ 3k þ 1 5 3 F k;6 ¼ k þ 4k þ 3k 6 4 2 F k;7 ¼ k þ 5k þ 6k þ 1 7 5 3 F k;8 ¼ k þ 6k þ 10k þ 4k S. Falcon, A´ . Plaza / Chaos, Solitons and Fractals 41 (2009) 497–504 499 Each Pisano period begins with a zero, the first of which corresponds to the remainder of F k;0 ¼ 0 (mod m) and always ends with an ‘‘1” as it will be shown bellow. The period-length corresponding to the value m is the number of remainders inside a period and it is denoted pkðmÞ. Since any period may be contained one or more zeros, the set of remainders between two consecutive zeros including the first zero, but not the second one will be called chain. Later on, it will be proved that all the chains have the same number of elements and that every Pisano period contains either 1, or 2 or 4 chains. Let us denote fF k;n mod mgn2N the sequence of the remainders of the k-Fibonacci sequence modulo m. Theorem 2. fF k;n mod mgn2N is a simple periodic sequence of period less than m2. See [25,26] for a proof for classical Fibonacci sequence. The same argument applies to k-Fibonacci numbers. For the sake of clarity in the exposition, we will write fr1; r2; ...g the sequence of remainders fF k;n mod mgn2N and we will consider the m2 pairs of the remainders ðr1; r2Þ; ðr2; r3Þ; ...ðrp2; rp2þ1Þ. If each one of these pairs is dif- ferent from any other, then one of them should be of the form ð0; 1Þ since there are only m2 pairs of numbers modulo m. If ðrp; rpþ1Þ¼ð0; 1Þ, then evidently, ðrpþ1; rpþ2Þ¼ð1; kÞ, and therefore the sequence of the remainders is periodic with period p. On the other hand, if two of these pairs are the same, for example, if ðri; riþ1Þ¼ðrj; rjþ1Þ for i < j, then also and ðriÀ1; riÞ¼ðrjÀ1; rjÞ and so on, until eventually ðr1; r2Þ¼ðrjÀi; rjÀiþ1Þ¼ð1; kÞ, and therefore the sequence is periodic with period j À i. Corollary 3. If m > 1 every Pisano period begins with 0; 1; ... Q ei ei ei Theorem 4. If the prime factorization of m is m ¼ pi , then pkðlcmðpi ÞÞ ¼ lcmðpk ðpi ÞÞ. ei ei Proof. Each pkðpi Þ is the period-length of the corresponding sequence fF k;n ðmod pi Þg, that is, the sequence ei ei fF k;m ðmod pi Þg repeats only after blocks of length cpkðpi Þ. On the other hand, pk ðmÞ is the period-length of the ei sequence fF k;n ðmod mÞ, which implies that fF k;n ðmod pi Þg repeats after pkðmÞ terms for all values of i, and since ei h any such number gives a period of fF k;n ðmod mÞ, we conclude that pkðmÞ¼lcmðpkðpi ÞÞ. Corollary 5. If rjm, then pkðrÞjpkðmÞ. Remark 6. p2rþ1ð2Þ¼3, and p2rð2Þ¼2, for r ¼ 1; 2; 3; ... Theorem 7. If the first two chains of the Pisano sequence fF k;m mod mg are 0; a1; a2; ...; arÀ2; arÀ1, and 0; b1; b2; ...; brÀ1, then b2iþ1 ¼ arð2iþ1Þ and b2i ¼ m À arÀ2i. Proof. Considering that F k;rþ1 ¼ kF k;r þ F k;rÀ1 and that 0 F k;r ðmod mÞ it is obtained that b1 F k;rþ1 ðmod mÞF k;rÀ1 ðmod mÞarÀ1. 2 F k;rþ2 F k;rÀ2 Now, since F k;rþ2 ¼ðk þ 2ÞF k;r À F k;rÀ2, then b2 ¼ remainder m ¼remainder m ¼ m À arÀ2. Finally, iterating previous reasoning the proof is done. h As a direct consequence of previous theorem, if in a sequence of remainders of k-Fibonacci sequences modulo m appear consecutively 1,0, then 1 is the last value of a Pisano period, and 0 is the beginning of the following period.
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