DOCSLIB.ORG

A Note on Generalized K-Pell Numbers and Their Determinantal Representation -.:: Natural Sciences Publishing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Note on Generalized K-Pell Numbers and Their Determinantal Representation -.:: Natural Sciences Publishing J. Ana. Num. Theor. 4, No. 2, 153-158 (2016) 153 Journal of Analysis & Number Theory An International Journal http://dx.doi.org/10.18576/jant/040211 A Note on Generalized k-Pell Numbers and Their Determinantal Representation Ahmet Oteles¸¨ 1,∗ and Mehmet Akbulak2 1 Department of Mathematics, Faculty of Education, Dicle University, TR-21280, Diyarbakır, Turkey 2 Department of Mathematics, Art and Science Faculty, Siirt University, TR-56100, Siirt, Turkey Received: 3 Jan. 2016, Revised: 9 Apr. 2016, Accepted: 11 Apr. 2016 Published online: 1 Jul. 2016 Abstract: In this paper, we investigate permanents of an n × n (0,1,2)-matrix by contraction method. We show that the permanent of the matrix is equal to the generalized k-Pell numbers. Keywords: Pell sequence, Permanent, Contraction of a matrix 1 Introduction where the summation extends over all permutations σ of the symmetric group Sn. The well-known Pell sequence {Pn} is defined by the Let A = [ai j] be an m × n real matrix with row vectors recurrence relation, for n > 2 α1,α2,...,αm. We say A is contractible on column (resp. P = 2P + P row) k if column (resp. row) k contains exactly two n n−1 n−2 nonzero entries. Suppose A is contractible on column k where P1 = 1 and P2 = 2. with aik 6= 0 6= a jk and i 6= j. Then the (m − 1) × (n − 1) In [1], the authors defined k sequences of the matrix Ai j:k obtained from A by replacing row i with k α α generalized order-k Pell numbers Pn as shown: a jk i + aik j and deleting row j and column k is called the contraction of A on column k relative to rows i and j. i i i i Pn = 2Pn−1 + Pn−2 + ... + Pn−k (1) If A is contractible on row k with aki 6= 0 6= ak j and T for n > 0 and1 ≤ i ≤ k, with initial conditions T i 6= j, then the matrix Ak:i j = Ai j:k is called the h i 1 if n = 1 − i, contraction of A on row k relative to columns i and j. We Pi = for 1 − k ≤ n ≤ 0, n 0 otherwise, say that A can be contracted to a matrix B if either B = A or there exist matrices A0,A1,...,At (t ≥ 1) such that i where Pn is the nth term of the ith sequence. The sequence A0 = A, At = B, and Ar is a contraction of Ar−1 for k Pn is reduced to the usual Pell sequence {Pn} for k = 2. r = 1,...,t. 4 4 In [2], Lee defined the following matrix For example, for i = 4, they obtained P−3 = 1, P−2 = P4 = P4 = 0 and gave first few terms of generalized order −1 0 10 ... 0 10 ... 00 4-Pell sequence as: 11 1 ... 1 0 ... 00 1,2,5,13,34,88,228,.... L (n,k) = 011 1 ... 10 ... 00 . k . The authors called Pn the generalized k-Pell number 000 0 ... 00 ... 11 for i = k in (1)[1]. The permanent of an n-square matrix A = [ai j] is (n,2) defined by and showed that perL = Ln−1 and also n (n,k) (k) (k) perL = l , where Ln and ln are respectively the perA = ∑ ∏aiσ(i) n−1 σεSn i=1 nth Lucas number and nth k-Lucas number. ∗ Corresponding author e-mail: [email protected] c 2016 NSP Natural Sciences Publishing Cor. 154 A. Oteles¸,¨ M. Akbulak: A note on generalized k-pell numbers In [3], Minc defined generalized Fibonacci numbers of as follows: order r as 1 1 −1 0 ········· 0 0, if n < 0, . 11 11 .. 1, if n = 0, f (n,r)= .. r 01 11 −1 . ∑ f (n − k,r), if n > 0. . k=1 . .. .. .. .. .. .. Hn = . Minc also defined the n × n super-diagonal (0,1)-matrix . .. .. .. .. 1 0 F (n,r) = [ f ] as in the following: . i j . .. .. .. .. −1 . 1 1 0 0 . .. .. .. 1 ······ ··· . 0 ············ 01 1 1 .. .. . 0 .. .. 0 and . 12 3 0 ········· 0 F (n,r)= . .. .. 1 (2) .. . 10 0 0 . . .. .. .. . 01 0 1 1 . . .. .. . 0 ········· 0 11 . .. 1 01 1 .. Kn = . . . .. .. .. .. .. 0 where . . .. .. .. .. 1 1, if i = j + 1and 1 ≤ j ≤ r − 1, . . .. .. .. 1 1, if 1 ≤ i ≤ n − r − 2 and i ≤ j ≤ i + r − 2, fi j = 0 ············ 0 10 1, if n − r + 3 ≤ i ≤ n and i ≤ j ≤ n, 0, otherwise. They showed that Then he proved that (n−2) perHn = perHn = Pn perF (n,r)= f (n,r − 1). and (n−2) In [4], the authors denoted the matrix F (n,r) in (2) perKn = perKn = Rn as F (n,k) and obtained permanent of this matrix, the same where Pn is nth Pell number, Rn is nth Perrin number, and result in [3, Theorem 2], by applying contraction to the (n−2) (n−2) matrix F (n,k). Hn and Kn is (n − 2)th contraction of the matrix Hn and K , respectively. In [5], Kilic defined the n × n super-diagonal (0,1,2)- n In this paper, we show that the permanent of the n × n matrix S(k,n) as: (0,1,2)-matrix S(k,n) equals to the (n + 1)th generalized k-Pell number by the contraction method. 2 1 ··· 1 0 ··· 0 . 12 1 ... 1 .. .. 01 2 1 ... 0 2 Generalized k-Pell numbers by contraction . method S(k,n)= . .. .. .. .. 1 (3) . . .. .. .. .. In this section, we use generalized order-k Pell sequence . .. .. .. k . 1 Pn given in (1) for i = k. For this sequence, it can be 0 ········· 0 12 written for n > k ≥ 2 k k k k k and proved that the permanent of S(k,n) equals to the P1−k = 1, P2−k = P3−k = ... = P−1 = P0 = 0 (n + 1)th generalized k-Pell number. In [6], Yilmaz and Bozkurt defined the n × n upper and k k k k Hessenberg matrix Hn = [hi j] and n × n matrix Kn = [ki j] Pn = 2Pn−1 + Pn−2 + ... + Pn−k. c 2016 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 4, No. 2, 153-158 (2016) / www.naturalspublishing.com/Journals.asp 155 First few terms of the generalized k-Pell numbers are column 1 such that follows in [1] as: 52 0 ······ 0 k k k k . P1 = 2P0 + P−1 + ... + P1−k = 1, 12 1 .. k k k k . P2 = 2P1 + P0 + ... + P2−k = 2 × 1 = 2, .. .. (n) 01 2 . k k k k T1 = , P3 = 2P2 + P1 + ... + P3 k = 2 × 2 + 1 = 5, . − . .. .. .. .. 0 k k k k P4 = 2P3 + P2 + ... + P4−k = 2 × 5 + 2 + 1 = 13, . . .. .. .. 1 Pk 2Pk Pk ... Pk 2 13 5 2 1 34,.... 5 = 4 + 3 + + 5−k = × + + + = 0 ······ 0 12 Then, some terms of this sequence can be given for some (1) (1) (n) k and n values in the following table: where t11 = 5 = P3 and t12 = 2 = P2. Also the matrix T1 can be contracted on column 1 such that n k 1 2 3 4 5 6 7 8 9 10 2 1 2 5 12 29 70 169 408 985 2378 12 5 0 ······ 0 . 3 1 2 5 13 33 84 214 545 1388 3535 .. 4 1 2 5 13 34 88 228 591 1532 3971 1 2 1 . 5 1 2 5 13 34 89 232 605 1578 4116 .. (n) 01 2 1 . T = , 2 . It is easily seen that from the above table, Pk is the . .. .. .. .. 0 n usual Pell sequence {P } for k = 2. . n . .. .. .. 1 The following Lemma is well-known from [7]. 0 ······ 0 12 Lemma 2.1. Let A be a nonnegative integral matrix of order n for n > 1 and let B be a contraction of A. Then (2) (2) where t11 = 12 = P4 and t12 = 5 = P3. Continuing this process, we obtain the rth contraction of T (n) as follows perA = perB. (4) Pr+2 Pr+1 0 ······ 0 . Let S(k,n) be the (k + 1)st (0,1,2)-matrix of order n 1 2 1 .. (n,k) given in (3). For convenience, we use the notation S .. F (n,k) (n) 0 1 21 . instead of S(k,n). If be the matrix as in (2) and In Tr = , . be the identity matrix of order n, then it is immediately . .. .. .. .. 0 (n,k) F (n,k) (n,k) seen that S = + In. Then the matrix S is . . .. .. .. 1 contractible on column 1 relative to rows 1 and 2. In particular, if k = 2, the matrix S(n,k) reduced to the 0 ······ 0 12 (n) tridiagonal Toeplitz matrix T = tridiagn (1,2,1) of order n as follows: for 2 ≤ r ≤ n − 2. So, we have 21 0 0 (n) Pn Pn−1 ······ Tn−2 = . (6) 12 1 0 0 1 2 ··· . 01 2 1 .. By combining (4)and (6), we get (n) T = . (5) . .. .. .. .. (n) (n) . 0 perT = perT . n−2 . .. .. .. = 2P + P . 1 n n−1 0 ······ 0 12 = Pn+1, and the proof is completed. Lemma 2.2. Let the matrix T (n) beasin (5). Then (n,k) (t) Lemma 2.3. Let St = si j be the tth contraction of the (n,k) h i (n) matrix S given in (3)and 1 ≤ t ≤ n − 3. Then perT = Pn+1, k Pt 2, if j = 1, where P be the n 1 st Pell number. + n+1 ( + ) t+1 (t) ∑ Pk, if 2 j k t, (n) (r) s = r ≤ ≤ − for k > t +2 1, j r 1 Proof.
Load more

Recommended publications
  • Prime Divisors in the Rationality Condition for Odd Perfect Numbers
    Aid#59330/Preprints/2019-09-10/www.mathjobs.org RFSC 04-01 Revised The Prime Divisors in the Rationality Condition for Odd Perfect Numbers Simon Davis Research Foundation of Southern California 8861 Villa La Jolla Drive #13595 La Jolla, CA 92037 Abstract. It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 4m+1 ℓ 2αi 1) i=1 qi to establish that there do not exist any odd integers with equality (4k+1)4m+2−1 between σ(N) and 2N. The existence of distinct prime divisors in the repunits 4k , 2α +1 Q q i −1 i , i = 1,...,ℓ, in σ(N) follows from a theorem on the primitive divisors of the Lucas qi−1 sequences and the square root of the product of 2(4k + 1), and the sequence of repunits will not be rational unless the primes are matched. Minimization of the number of prime divisors in σ(N) yields an infinite set of repunits of increasing magnitude or prime equations with no integer solutions. MSC: 11D61, 11K65 Keywords: prime divisors, rationality condition 1. Introduction While even perfect numbers were known to be given by 2p−1(2p − 1), for 2p − 1 prime, the universality of this result led to the the problem of characterizing any other possible types of perfect numbers. It was suggested initially by Descartes that it was not likely that odd integers could be perfect numbers [13]. After the work of de Bessy [3], Euler proved σ(N) that the condition = 2, where σ(N) = d|N d is the sum-of-divisors function, N d integer 4m+1 2α1 2αℓ restricted odd integers to have the form (4kP+ 1) q1 ...qℓ , with 4k + 1, q1,...,qℓ prime [18], and further, that there might exist no set of prime bases such that the perfect number condition was satisfied.
    [Show full text]
  • ON SOME GAUSSIAN PELL and PELL-LUCAS NUMBERS Abstract
    Ordu Üniv. Bil. Tek. Derg., Cilt:6, Sayı:1, 2016,8-18/Ordu Univ. J. Sci. Tech., Vol:6, No:1,2016,8-18 ON SOME GAUSSIAN PELL AND PELL-LUCAS NUMBERS Serpil Halıcı*1 Sinan Öz2 1Pamukkale Uni., Science and Arts Faculty,Dept. of Math., KınıklıCampus, Denizli, Turkey 2Pamukkale Uni., Science and Arts Faculty,Dept. of Math., Denizli, Turkey Abstract In this study, we consider firstly the generalized Gaussian Fibonacci and Lucas sequences. Then we define the Gaussian Pell and Gaussian Pell-Lucas sequences. We give the generating functions and Binet formulas of Gaussian Pell and Gaussian Pell- Lucas sequences. Moreover, we obtain some important identities involving the Gaussian Pell and Pell-Lucas numbers. Keywords. Recurrence Relation, Fibonacci numbers, Gaussian Pell and Pell-Lucas numbers. Özet Bu çalışmada, önce genelleştirilmiş Gaussian Fibonacci ve Lucas dizilerini dikkate aldık. Sonra, Gaussian Pell ve Gaussian Pell-Lucas dizilerini tanımladık. Gaussian Pell ve Gaussian Pell-Lucas dizilerinin Binet formüllerini ve üreteç fonksiyonlarını verdik. Üstelik, Gaussian Pell ve Gaussian Pell-Lucas sayılarını içeren bazı önemli özdeşlikler elde ettik. AMS Classification. 11B37, 11B39.1 * [email protected], 8 S. Halıcı, S. Öz 1. INTRODUCTION From (Horadam 1961; Horadam 1963) it is well known Generalized Fibonacci sequence {푈푛}, 푈푛+1 = 푝푈푛 + 푞푈푛−1 , 푈0 = 0 푎푛푑 푈1 = 1, and generalized Lucas sequence {푉푛} are defined by 푉푛+1 = 푝푉푛 + 푞푉푛−1 , 푉0 = 2 푎푛푑 푉1 = 푝, where 푝 and 푞 are nonzero real numbers and 푛 ≥ 1. For 푝 = 푞 = 1, we have classical Fibonacci and Lucas sequences. For 푝 = 2, 푞 = 1, we have Pell and Pell- Lucas sequences.
    [Show full text]
  • Arxiv:1206.0407V3 [Math.NT] 22 Mar 2013 N Egto Utpehroi U.B Ovnin Eput We Convention, by Sum
    NEW PROPERTIES OF MULTIPLE HARMONIC SUMS MODULO p AND p-ANALOGUES OF LESHCHINER’S SERIES KH. HESSAMI PILEHROOD, T. HESSAMI PILEHROOD, AND R. TAURASO Abstract. In this paper we present some new binomial identities for multiple harmonic sums whose indices are the sequences ({1}a, c, {1}b), ({2}a, c, {2}b) and prove a number of congruences for these sums modulo a prime p. The congruences obtained allow us to find nice p-analogues of Leshchiner’s series for zeta values and to refine a result due to M. Hoffman and J. Zhao about the set of generators of the multiple harmonic sums of weight 7 and 9 modulo p. ∗ As a further application we provide a new proof of Zagier’s formula for ζ ({2}a, 3, {2}b) based on a finite identity for partial sums of the zeta-star series. 1. Introduction In the last few years there has been a growing attention in the study of p-adic analogues of various binomial series related to multiple zeta values, which are nested generalizations of s s the classical Riemann zeta function ζ(s) = n=1 1/n . The main reason of interest of such p-analogues is that they are related to divisibility properties of multiple harmonic sums which can be considered as elementary “bricks” forP expressing complicated congruences. Before discussing this further we recall the precise definitions of such objects. ∗ r For r ∈ N, s = (s1,s2,...,sr) ∈ (Z ) , and a non-negative integer n, the alternating multiple harmonic sum is defined by r ki sgn (si) Hn(s1,s2,...,sr)= k|si| 1≤k1<kX2<...<kr≤n Yi=1 i and the “odd” alternating multiple harmonic sum is given by r sgn (s )ki H (s ,s ,...,s )= i .
    [Show full text]
  • And Its Properties on the Sequence Related to Lucas Numbers
    Mathematica Aeterna, Vol. 2, 2012, no. 1, 63 - 75 On the sequence related to Lucas numbers and its properties Cennet Bolat Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Ahmet I˙pek Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Hasan Köse Department of Mathematics, Science Faculty, Selcuk University 42031, Konya,Turkey Abstract The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recur- rence relation. In this article, we study a new generalization {Lk,n}, with initial conditions Lk,0 = 2 and Lk,1 = 1, which is generated by the recurrence relation Lk,n = kLk,n−1 + Lk,n−2 for n ≥ 2, where k is integer number. Some well-known sequence are special case of this generalization. The Lucas sequence is a special case of {Lk,n} with k = 1. Modified Pell-Lucas sequence is {Lk,n} with k = 2. We produce an extended Binet’s formula for {Lk,n} and, thereby, identities such as Cassini’s, Catalan’s, d’Ocagne’s, etc. using matrix algebra. Moreover, we present sum formulas concerning this new generalization. Mathematics Subject Classi…cation: 11B39, 15A23. Keywords: Classic Fibonacci numbers; Classic Lucas numbers; k-Fibonacci numbers; k-Lucas numbers, Matrix algebra. 64 C. Bolat, A. Ipek and H. Kose 1 Introduction In recent years, many interesting properties of classic Fibonacci numbers, clas- sic Lucas numbers and their generalizations have been shown by researchers and applied to almost every …eld of science and art.
    [Show full text]
  • Adopting Number Sequences for Shielding Information
    Manish Bansal et al, International Journal of Computer Science and Mobile Computing, Vol.3 Issue.11, November- 2014, pg. 660-667 Available Online at www.ijcsmc.com International Journal of Computer Science and Mobile Computing A Monthly Journal of Computer Science and Information Technology ISSN 2320–088X IJCSMC, Vol. 3, Issue. 11, November 2014, pg.660 – 667 RESEARCH ARTICLE Adopting Number Sequences for Shielding Information Mrs. Sandhya Maitra, Mr. Manish Bansal, Ms. Preety Gupta Associate Professor - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India Student - Department of Computer Applications and Institute of Information Technology and Management, India [email protected] , [email protected] , [email protected] Abstract:-The advancement of technology and global communication networks puts up the question of safety of conveyed data and saved data over these media. Cryptography is the most efficient and feasible mode to transfer security services and also Cryptography is becoming effective tool in numerous applications for information security. This paper studies the shielding of information with the help of cryptographic function and number sequences. The efficiency of the given method is examined, which assures upgraded cryptographic services in physical signal and it is also agile and clear for realization. 1. Introduction Cryptography is the branch of information security. The basic goal of cryptography is to make communication secure in the presence of third party (intruder). Encryption is the mechanism of transforming information into unreadable (cipher) form. This is accomplished for data integrity, forward secrecy, authentication and mainly for data confidentiality.
    [Show full text]
  • On the Connections Between Pell Numbers and Fibonacci P-Numbers
    Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 27, 2021, No. 1, 148–160 DOI: 10.7546/nntdm.2021.27.1.148-160 On the connections between Pell numbers and Fibonacci p-numbers Anthony G. Shannon1, Ozg¨ ur¨ Erdag˘2 and Om¨ ur¨ Deveci3 1 Warrane College, University of New South Wales Kensington, Australia e-mail: [email protected] 2 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] 3 Department of Mathematics, Faculty of Science and Letters Kafkas University 36100, Turkey e-mail: [email protected] Received: 24 April 2020 Revised: 4 January 2021 Accepted: 7 January 2021 Abstract: In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices. Keywords: Pell sequence, Fibonacci p-sequence, Matrix, Representation. 2010 Mathematics Subject Classification: 11K31, 11C20, 15A15. 1 Introduction The well-known Pell sequence fPng is defined by the following recurrence relation: Pn+2 = 2Pn+1 + Pn for n ≥ 0 in which P0 = 0 and P1 = 1. 148 There are many important generalizations of the Fibonacci sequence. The Fibonacci p-sequence [22, 23] is one of them: Fp (n) = Fp (n − 1) + Fp (n − p − 1) for p = 1; 2; 3;::: and n > p in which Fp (0) = 0, Fp (1) = ··· = Fp (p) = 1.
    [Show full text]
  • Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and Their Explicit Formulas by Ian Walker Ju
    Explorations in Recursion with John Pell and the Pell Sequence Recurrence Relations and their Explicit Formulas By Ian Walker June 2011 MST 501 Research Project In partial fulfillment for the Master’s of Teaching Mathematics (M.S.T.) Portland State University, Portland, Oregon John Pell (1611-1685) 2 John Pell (1611-1685) An “obscure” English Mathematician • Part of the 17th century intellectual history of England and of Continental Europe. • Pell was married with eight children, taught math at the Gymnasium in Amsterdam, and was Oliver Cromwell’s envoy to Switzerland. • Pell was well read in classical and contemporary mathematics. • Pell had correspondence with Descartes, Leibniz, Cavendish, Mersenne, Hartlib, Collins and others. • His main mathematical focus was on mathematical tables: tables of squares, sums of squares, primes and composites, constant differences, logarithms, antilogarithms, trigonometric functions, etc. 3 John Pell (1611-1685) An “obscure” English Mathematician • Many of Pell’s booklets of tables and other works do not list himself as the author. • Did not publish much mathematical work. Is more known for his activities, correspondence and contacts. • Only one of his tables was ever published (1672), which had tables of the first 10,000 square numbers. • His best known published work is, “An Introduction to Algebra”. It explains how to simplify and solve equations. • Pell is credited with the modern day division symbol and the double-angle tangent formula. • Pell is best known, only by name, for the Pell Sequence and the Pell Equation. 4 John Pell (1611-1685) An “obscure” English Mathematician • Division Symbol: • Double-Angle Tangent Formula: 2tan tan2 2 • Pell Sequence: 1 tan p 2p p p 1, p 2,n 2 n n1 n2 0 1 • Pell Equation: 2 2 x 2y 1 5 John Pell (1611-1685) An “obscure” English Mathematician • Both the Pell Sequence and the Pell Equation are erroneously named after him.
    [Show full text]
  • Arxiv:1606.08690V5 [Math.NT] 27 Apr 2021 on Prime Factors of Mersenne
    On prime factors of Mersenne numbers Ady Cambraia Jr,∗ Michael P. Knapp,† Ab´ılio Lemos∗, B. K. Moriya∗ and Paulo H. A. Rodrigues‡ [email protected] [email protected] [email protected] [email protected] paulo [email protected] April 29, 2021 Abstract n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1. Let ω(n) be the number of distinct prime divisors of n. In this short note, we present a description of the Mersenne numbers satisfying ω(Mn) ≤ 3. Moreover, we prove that the inequality, (1−ǫ) log log n given ǫ> 0, ω(Mn) > 2 − 3 holds for almost all positive integers n. Besides, a we present the integer solutions (m, n, a) of the equation Mm+Mn = 2p with m,n ≥ 2, p an odd prime number and a a positive integer. 2010 Mathematics Subject Classification: 11A99, 11K65, 11A41. Keywords: Mersenne numbers, arithmetic functions, prime divisors. 1 Introduction arXiv:1606.08690v5 [math.NT] 27 Apr 2021 n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1, for n ≥ 0. A simple argument shows that if Mn is a prime number, then n is a prime number. When Mn is a prime number, it is called a Mersenne prime. Throughout history, many researchers sought to find Mersenne primes. Some tools are very important for the search for Mersenne primes, mainly the Lucas-Lehmer test. There are papers (see for example [1, 5, 21]) that seek to describe the prime factors of Mn, where Mn is a composite number and n is a prime number.
    [Show full text]
  • Repdigits As Sums of Two Lucas Numbers∗
    Applied Mathematics E-Notes, 20(2020), 33-38 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ Repdigits As Sums Of Two Lucas Numbers Zafer ¸Siary, Refik Keskinz Received 11 Feburary 2019 Abstract Let (Ln) be the Lucas sequence defined by Ln = Ln 1 +Ln 2 for n 2 with initial conditions L0 = 2 ≥ and L1 = 1. A repdigit is a nonnegative integer whose digits are all equal. In this paper, we show that if Ln + Lm is a repdigit, then Ln + Lm = 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 77, 333. 1 Introduction Let (Fn)n 0 be the Fibonacci sequence satisfying the recurrence relation Fn+2 = Fn+1 + Fn with initial ≥ conditions F0 = 0 and F1 = 1. Let (Ln)n 0 be the Lucas sequence following the same recursive pattern as ≥ th the Fibonacci sequence, but with initial conditions L0 = 2 and L1 = 1.Fn and Ln are called n Fibonacci number and nth Lucas number, respectively. It is well known that n n F = and L = n + n, (1) n n where 1 + p5 1 p5 = and = , 2 2 which are the roots of the characteristic equation x2 x 1 = 0. Also, the following relation between nth Lucas number Ln and is well known: n 1 n Ln 2 (2) ≤ ≤ for n 0. The inequality (2) can be proved by induction. A≥ repdigit is a nonnegative integer whose digits are all equal. Recently, some mathematicians have investigated the repdigits which are sums or products of any number of Fibonacci numbers, Lucas numbers, and Pell numbers.
    [Show full text]
  • Fermat Padovan and Perrin Numbers
    1 2 Journal of Integer Sequences, Vol. 23 (2020), 3 Article 20.6.2 47 6 23 11 Fermat Padovan And Perrin Numbers Salah Eddine Rihane Department of Mathematics Institute of Science and Technology University Center of Mila Algeria [email protected] Ch`efiath Awero Adegbindin Institut de Math´ematiques et de Sciences Physiques Dangbo B´enin [email protected] Alain Togb´e Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S., U.S. 421 Westville, IN 46391 USA [email protected] Abstract In this paper, we determine all the Padovan and Perrin numbers that are also Fermat numbers. 1 1 Introduction The Padovan sequence Pm m 0 is defined by { } ≥ Pm+3 = Pm+1 + Pm, (1) for m 0, where P0 = P1 = P2 = 1. This is the sequence A000931 in the On-Line Encyclopedia≥ of Integer Sequences (OEIS). A few terms of this sequence are 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, ··· Let Em m 0 be the Perrin sequence given by { } ≥ Em+3 = Em+1 + Em, (2) for m 0, where E = 3, E = 0, and E = 2. Its first few terms are ≥ 0 1 2 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367, 486, 644, 853, ··· It is the sequence A001608 in the OEIS. Let us also recall that a Fermat number is a number of the form m =22 +1, Fm where m is a nonnegative integer.
    [Show full text]
  • Repdigits As Sums of Two Generalized Lucas Numbers
    Turkish Journal of Mathematics Turk J Math (2021) 45: 1166 – 1179 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-2011-59 Repdigits as sums of two generalized Lucas numbers Sai Gopal RAYAGURU1;∗, Jhon Jairo BRAVO2 1Department of Mathematics, National Institute of Technology Rourkela, Odisha, India 2Department of Mathematics, University of Cauca, Popayán, Colombia Received: 18.11.2020 • Accepted/Published Online: 20.03.2021 • Final Version: 20.05.2021 Abstract: A generalization of the well–known Lucas sequence is the k -Lucas sequence with some fixed integer k ≥ 2. The first k terms of this sequence are 0;:::; 0; 2; 1, and each term afterwards is the sum of the preceding k terms. In this paper, we determine all repdigits, which are expressible as sums of two k -Lucas numbers. This work generalizes a prior result of Şiar and Keskin who dealt with the above problem for the particular case of Lucas numbers and a result of Bravo and Luca who searched for repdigits that are k -Lucas numbers. Key words: Generalized Lucas number, repdigit, linear form in logarithms, reduction method 1. Introduction For an integer k ≥ 2, let the k -generalized Fibonacci sequence (or simply, the k -Fibonacci sequence) F (k) := (k) (Fn )n≥2−k be defined by the linear recurrence sequence of order k (k) (k) (k) ··· (k) ≥ Fn = Fn−1 + Fn−2 + + Fn−k for all n 2; (k) (k) ··· (k) (k) with initial conditions F−(k−2) = F−(k−3) = = F0 = 0 and F1 = 1. The above sequence is one among the several generalizations of the Fibonacci sequence (Fn)n≥0 .
    [Show full text]
  • Arxiv:2001.11839V1 [Math.CO] 30 Jan 2020 Ihiiilvalues Initial with Ubr Aebcm N Ftems Oua Eune Ostudy by to Denoted Sequences Numbers, Combinatorics
    SOME RESULTS ON AVERAGE OF FIBONACCI AND LUCAS SEQUENCES DANIEL YAQUBI AND AMIRALI FATEHIZADEH Abstract. The numerical sequence in which the n-th term is the average (that is, arithmetic mean) of the of the first n Fibonacci numbers may be found in the OEIS (see A111035 ). An interesting question one might pose is which terms of the sequences n ∞ n ∞ 1 1 Fi and Li (0.1) n i=1 n=1 n i=1 n=1 X X are integers? The average of the first three Fibonacci sequence, (1 + 1 + 3)/3=4/3, is not whereas the 24 Pi=1 Fi average of the first 24 Fibonacci sequence, 24 = 5058 is. In this paper, we address this question and also present some properties of average Fibonacci and Lucas numbers using the Wall-Sun-Sun prime conjecture. 1. Introduction Fibonacci numbers originally arose in a problem in Liber Abaci, published in 1202, which was one of the first texts to describe the Hindu-Arabic numeral system. Since then Fibonacci numbers have become one of the most popular sequences to study, appearing in a wealth of problems not only in enumerative combinatorics. The Fibonacci numbers, denoted by F ∞ , are defined by the following recurrence relation { n}n=0 Fn+1 = Fn + Fn 1, (1.1) − with initial values F0 = 0 and F1 = 1 . The first elements of this sequence are given in (A000045 ), as 0 1 1 2 3 5 8 13 21 .... arXiv:2001.11839v1 [math.CO] 30 Jan 2020 The Fibonacci numbers are closely related to binomial coefficients; it is a well-known fact that they are given by the sums of the rising diagonal lines of Pascal’s triangle (see [1]) n ⌊ 2 ⌋ n i F = − .
    [Show full text]