DOCSLIB.ORG

Proceedings of the Sixteenth International Conference on Fibonacci Numbers and Their Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Proceedings of the Sixteenth International Conference on Fibonacci Numbers and Their Applications Proceedings of the Sixteenth International Conference on Fibonacci Numbers and Their Applications Rochester Institute of Technology, Rochester, New York, 2014 July 20{27 edited by Peter G. Anderson Rochester Institute of Technology Rochester, New York Christian Ballot Universit´eof Caen-Basse Normandie Caen, France William Webb Washington State University Pullman, Washington ii The Sixteenth Conference The 16th International Conference on Fibonacci Numbers and their Applica- tions was held on the large campus of the Rochester Institute of Technology, situated several miles off from downtown. It hosted about 65 participants from at least a dozen countries and all continents, northern Americans being most represented. Besides regular and occasional participants, there were a number of people who attended this conference for the first time. For instance, M´arton, 24, from Hungary, took three flights to reach Rochester; it was his first flying experiences, and we believe many appreciated his presence, and he himself en- joyed the whole package of the conference. These conferences are very congenial, being both scientific, social, and cultural events. This one had the peculiarity of having three exceptional presentations open to the public held on the Wednesday morning in a large auditorium filled with local young people and students, in addition to the conference participants. The Edouard´ Lucas invited lecturer, Jeffrey Lagarias, gave a broad well-applauded historic talk which ran from antiquity to present; Larry Ericksen, painter and mathematician, also had us travel through time and space, commenting on often famous artwork|okay, maybe the Golden ratio appeared a few too many times|and Arthur Benjamin, mathemagician (and mathematician) who, for some of his magics, managed the feat of both performing and explaining without loosing the audience a second. Peter Anderson was the grand host and organizer|at least two of the edi- tors wish to express their thanks to him|but many among the 65 participants will also remember fondly Ginny Gross-Abbey and Kim Shearer, for their un- compromising week-long help performed with competence and joy. We also offer our sincere thank you to the Rochester Institute of Technology, the B. Thomas Golisano College of Computing and Information Sciences, and its Dean Andrew Sears for generously hosting our meeting and providing us their superb facilities. A wine and cheese reception was held on the Sunday evening, which for many was their arrival date, an optional, memorable cruise and dinner on the Erie Canal was organized on the Tuesday; Wednesday afternoon was off with the possibility of visiting George Eastman's house, now an impressive museum. The conference banquet was held on the Thursday at a most original place, Artisan Works, surrounded by fine art. A day-trip to Niagara Falls took place on the last Saturday. iii iv Forword The sixteenth International Conference on Fibonacci Numbers and their Appli- cations was held in Rochester, New York, USA, July 20-26, on the campus of the Rochester Institute of Technology. This book contains 23 items which include a beautiful paper of Marjorie Bicknell-Johnson on the 50+ years of the existence of the Fibonacci Association, a compendium of problems posed, with occasional solutions, or partial solutions, found during the problem sessions and put together by Clark Kimberling, and 21 research articles from among the 49 papers and abstracts presented at the conference. These articles were selected and criticized by expert referees, whose time and care brought added value to this volume. Even though Fibonacci numbers and recurrences are the common bond to them all, the variety of topics and creativity of the papers compiled herein, is a testimony to the liveliness of this area of mathematics. It is our belief that the investigations reported in these proceedings, the 15th such proceedings emanating from this international conference, will stir up the curiosity of a number of researchers. Peter G. Anderson Christian Ballot William Webb v vi Contents Marjorie Bicknell-Johnson The Fibonacci Association: Historical Snapshots . 1 Clark Kimberling, Problem Editor Problem Proposals .......................... 5 Peter G. Anderson More Properties of the Zeckendorf Array . 15 Krassimir T. Atanassov, Daryl R. DeFord, and Anthony G. Shannon Pulsated Fibonacci Recurrences ................... 22 Arthur T. Benjamin and Elizabeth Reiland Combinatorial Proofs of Fibonomial Identities . 28 Andrew Best, Patrick Dynes, Xixi Edelsbrunner, Brian McDonald, Steven J. Miller, Kimsy Tor, Caroline Turnage-Butterbaugh, and Madeleine Weinstein Benford Behavior of Zeckendorf Decompositions . 35 Andrew Best, Patrick Dynes, Xixi Edelsbrunner, Brian McDonald, Steven J. Miller, Kimsy Tor, Caroline Turnage-Butterbaugh, and Madeleine Weinstein Gaussian Behavior of the Number of Summands in Zeckendorf Decompositions in Small Intervals . 47 Daniel Birmajer, Juan B. Gil, and Michael D. Weiner Convolutions of Tribonacci, Fuss-Catalan, and Motzkin Sequences 54 Jens-P. Bode and Heiko Harborth Steinhaus Triangles with Generalized Pascal Addition . 61 Minerva Catral, Pari Ford, Pamela Harris, Steven J. Miller, and Dawn Nelson Generalizing Zeckendorf's Theorem: The Kentucky Sequence . 68 Curtis Cooper Algebraic Statements Similar to Those in Ramanujan's \Lost Note- book" ................................. 91 Daryl DeFord Enumerating Distinct Chessboard Tilings . 102 M. Cetin Firengiz and Naim Tuglu On the q-Seidel Matrix . 117 Russell Jay Hendel A Cayley-Hamilton and Circulant Approach to Jump Sums . 124 vii Clark Kimberling and Peter J. C. Moses The Infinite Fibonacci Tree and Other Trees Generated by Rules 136 Takao Komatsu, Zuzana Mas´akov´a,and Edita Pelantov´a Higher-Order Identities for Fibonacci Numbers . 150 Florian Luca and Pantelimon St˘anic˘a On Fibonacci Numbers which are Elliptic Korselt Numbers . 164 Sam Northshield Three Analogues of Stern's Diatomic Sequence . 168 G. K. Panda and A. K. Panda Balancing-like Sequences Associated with Integral Standard Devi- ations of Consecutive Natural Numbers . 187 Pantelimon St˘anic˘a Normic Continued Fractions in Totally and Tamely Ramified Ex- tensions of Local Fields . 193 William Webb and Nathan Hamlin Compositions and Recurrences . 201 Paul Thomas Young Symmetries of Stirling Number Series . 205 Yifan Zhang and George Grossman Diophantine Triples and the Extendibility of 1; 2; 5 ; 1; 5; 10 . 212 Conference Participants f g f g . ....................................216 viii THE FIBONACCI ASSOCIATION: HISTORICAL SNAPSHOTS MARJORIE BICKNELL-JOHNSON Abstract. The Fibonacci Quarterly is now in its 52nd year of publication. V. E. Hoggatt, Jr., was editor for 18 years; G. E. Bergum, also editor for 18 years; and current editor, Curtis Cooper, 1998 to date. Marjorie Bicknell-Johnson was secretary of the Fibonacci Association from its beginning until 2010. This article gives a short history of the Fibonacci Association and some vignettes to bring that history to life. 1. Historical Snapshots and Rambles The first Fibonacci Quarterly, published in February 1963, had a subscription rate of $4.00 per year, and its Editor, Verner E. Hoggatt, Jr., held that position for L7 years. Vern’s friends told him the Quarterly wouldn’t last three years. Undaunted, he kept a mental list of “backsliders” who had not renewed their subscriptions and contacted each one personally. He persuaded, cajoled, and implored them so much that, in the end, it was impossible to say no. Vern corresponded and made friends with mathematicians from all over the world. He once hosted the world-famous mathematician Paul Erd¨osfor a month. Erd¨osarrived with one small suitcase, filled with silk underwear (because of allergies he was said to have). I met him, but he much preferred to talk to Vern. I found him strange; he had no home, no family, and all of his possessions were in his suitcase. I think he found me strange as well, visiting a college professor with my two little “epsilons,” the Erd¨osword for children. Erd¨osand Vern worked on dozens of problems during that time and wrote one paper [6] together, which I typed. As many of you may know, it was considered a great honor to have published a paper with Erd¨os, so much so that on Google, one can find the Erd¨osnumber of those who wrote with Erd¨os: 1; those who wrote with someone who wrote with Erd¨os:2; and so on. So, Vern has Erd¨os number 1, while Jerry Bergum and I each have Erd¨osnumber 2. The Managing Editor, Brother Alfred Brousseau, who was equally positive and enthusiastic about anything dealing with Fibonacci numbers, typed the first issue of the Quarterly and kept track of subscriptions and the bank account. He played the accordion and loved to lead group singing. In tune with his personality, he wrote the ballad, “Do What Comes Fibonaturally,” to the melody of “The Blue-Tail Fly.” Additionally, he compiled a bibliography of 700 Fibonacci references ranging from recreational to serious research, quite a feat in pre-computer times, and collected cones from every species of pine found in California to illustrate the Fibonacci patterns found in spirals of their scales. He inspired Vern to grow a large sunflower, his so-called Lucas sunflower, which had 76 clockwise spirals and 47 counterclockwise ones. As one of Professor Hoggatt’s students, I came on board in 1962. He was well liked by students at San Jose State College; they nicknamed him “Professor Fibonacci,” and I soon found out why: he took any and every opportunity to lecture on the Fibonacci sequence. We all loved it. 1 THE FIBONACCI QUARTERLY Dr. Hoggatt—I called him that as his student—brought unusual problems to us for home- work. A curiosa in Scripta Mathematica [7] claimed that, if the nine positive digits are ar- ranged in a square array, the non-negative determinant values range from 0 through 512 but with a couple dozen missing values. So we were each assigned to write and evaluate 20 such determinants—find those missing values, or show that they were indeed impossible.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Integer Sequences
    UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular
    [Show full text]
  • Padovan-Like Sequences and Generalized Pascal's Triangles
    An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si.Mat. (N.S.) Tomul LXVI, 2020, f. 1 Padovan-like sequences and generalized Pascal’s triangles Giuseppina Anatriello Giovanni Vincenzi · Abstract The Padovan sequence and the Plastic number are mathematical objects of central interest among architects, and from this point of view, their importance is com- parable to that of the Fibonacci sequence and the Golden mean. Padovan-like sequences, such as Perrin sequence or the Cordonnier sequence, also have mathematical applications. In this article, we will characterize a large family of Padovan-like sequences in terms of ‘Padovan diagonals’ of generalized Pascal’s triangles. Keywords Padovan-like sequences Plastic number Combinatorial identities Kepler limit Generalized Pascal’s triangle · Computational methods· · · · Mathematics Subject Classification (2010) 05A19 03D20 00A67 11Y55 · · · 1 Introduction The Fibonacci sequence, which we will denote by (Fn), is defined by F0 = 0,F1 = 1 Fn = Fn 1 + Fn 2, n > 1, (1.1) − − and it has been of wide interest since his first appearance in the book Liber Abaci published in 1202. Four centuries later, Kepler showed (see the book “Harmonices Mundi”, 1619, p. 273, see [20], [21]) that: F lim n+1 = Φ (1.2) n Fn where Φ denotes the highly celebrated Golden mean, which appears in sev- eral human fields, especially in Art and Architecture, even if sometimes not always appropriately (see e.g. [14], [24] and references therein). Among the most relevant generalizations of the Fibonacci sequence, there is the Tribonacci sequence (Tn). It is a third order recursive sequence defined by T0 = 0,T1 = T2 = 1, and Tn = Tn 1 + Tn 2 + Tn 3, n > 1.
    [Show full text]
  • Complete Generalized Fibonacci Sequences Modulo Primes
    Complete Generalized Fibonacci Sequences Modulo Primes Mohammad Javaheri, Nikolai A. Krylov Siena College, Department of Mathematics 515 Loudon Road, Loudonville NY 12211, USA [email protected], [email protected] Abstract We study generalized Fibonacci sequences F = P F QF − n+1 n − n 1 with initial values F0 = 0 and F1 = 1. Let P,Q be nonzero integers such that P 2 4Q is not a perfect square. We show that if Q = 1 − ∞ ± then the sequence F misses a congruence class modulo every { n}n=0 prime large enough. On the other hand, if Q = 1, we prove that ∞ 6 ± (under GRH) the sequence F hits every congruence class mod- { n}n=0 ulo infinitely many primes. A generalized Fibonacci sequence with integer parameters (P, Q) is a second order homogeneous difference equation generated by F = P F QF − , (1) n+1 n − n 1 n 1, with initial values F = 0 and F = 1; we denote this sequence by ≥ 0 1 F = [P, Q]. In this paper, we are interested in studying the completeness arXiv:1812.01048v1 [math.NT] 3 Dec 2018 problem for generalized Fibonacci sequences modulo primes. Definition 1. A sequence F ∞ is said to be complete modulo a prime p, { n}n=0 if the map e : N0 Zp = Z/pZ defined by e(n)= Fn + pZ is onto. In other words, the sequence→ is complete modulo p iff k Z n 0 F k (mod p). ∀ ∈ ∃ ≥ n ≡ 1 The study of completeness problem for the Fibonacci sequence (with pa- rameters (1, 1)) began with Shah [8] who showed that the Fibonacci se- − quence is not complete modulo primes p 1, 9 (mod 10).
    [Show full text]
  • Plastic Number: Construction and Applications
    SA 201 AR 2 - - A E d C v N a E n c R e E d F R N e Advanced Research in Scientific Areas 2012 O s C e a L r A c h U T i n R I S V c - i e s n a t e i r f i c A December, 3. - 7. 2012 Plastic Number: Construction and Applications Luka Marohnić Tihana Strmečki Polytechnic of Zagreb, Polytechnic of Zagreb, 10000 Zagreb, Croatia 10000 Zagreb, Croatia [email protected] [email protected] Abstract—In this article we will construct plastic number in a heuristic way, explaining its relation to human perception in three-dimensional space through architectural style of Dom Hans van der Laan. Furthermore, it will be shown that the plastic number and the golden ratio special cases of more general defini- tion. Finally, we will explain how van der Laan’s discovery re- lates to perception in pitch space, how to define and tune Padovan intervals and, subsequently, how to construct chromatic scale temperament using the plastic number. (Abstract) Keywords-plastic number; Padovan sequence; golden ratio; music interval; music tuning ; I. INTRODUCTION In 1928, shortly after abandoning his architectural studies and becoming a novice monk, Hans van der Laan1 discovered a new, unique system of architectural proportions. Its construc- tion is completely based on a single irrational value which he 2 called the plastic number (also known as the plastic constant): 4 1.324718... (1) Figure 1. Twenty different cubes, from above 3 This number was originally studied by G.
    [Show full text]
  • Modular Periodicity of Exponential Sums of Symmetric Boolean Functions and Some of Its Consequences
    MODULAR PERIODICITY OF EXPONENTIAL SUMS OF SYMMETRIC BOOLEAN FUNCTIONS AND SOME OF ITS CONSEQUENCES FRANCIS N. CASTRO AND LUIS A. MEDINA Abstract. This work brings techniques from the theory of recurrent integer sequences to the problem of balancedness of symmetric Booleans functions. In particular, the periodicity modulo p (p odd prime) of exponential sums of symmetric Boolean functions is considered. Periods modulo p, bounds for periods and relations between them are obtained for these exponential sums. The concept of avoiding primes is also introduced. This concept and the bounds presented in this work are used to show that some classes of symmetric Boolean functions are not balanced. In particular, every elementary symmetric Boolean function of degree not a power of 2 and less than 2048 is not balanced. For instance, the elementary symmetric Boolean function in n variables of degree 1292 is not balanced because the prime p = 176129 does not divide its exponential sum for any positive integer n. It is showed that for some symmetric Boolean functions, the set of primes avoided by the sequence of exponential sums contains a subset that has positive density within the set of primes. Finally, in the last section, a brief study for the set of primes that divide some term of the sequence of exponential sums is presented. 1. Introduction Boolean functions are beautiful combinatorial objects with applications to many areas of mathematics as well as outside the field. Some examples include combinatorics, electrical engineering, game theory, the theory of error-correcting codes, and cryptography. In the modern era, efficient implementations of Boolean functions with many variables is a challenging problem due to memory restrictions of current technology.
    [Show full text]
  • On the Parity of the Order of Appearance in the Fibonacci Sequence
    mathematics Article On the Parity of the Order of Appearance in the Fibonacci Sequence Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic; [email protected]; Tel.: +420-49-333-2860 Abstract: Let (Fn)n≥0 be the Fibonacci sequence. The order of appearance function (in the Fibonacci sequence) z : Z≥1 ! Z≥1 is defined as z(n) := minfk ≥ 1 : Fk ≡ 0 (mod n)g. In this paper, among other things, we prove that z(n) is an even number for almost all positive integers n (i.e., the set of such n has natural density equal to 1). Keywords: order of appearance; Fibonacci numbers; parity; natural density; prime numbers 1. Introduction Let (Fn)n be the sequence of Fibonacci numbers which is defined by the recurrence Fn+2 = Fn+1 + Fn, with initial values F0 = 0 and F1 = 1. For any integer n ≥ 1, the order of apparition (or the rank of appearance) of n (in the Fibonacci sequence), denoted by z(n), is the positive index of the smallest Fibonacci number which is a multiple of n (more precisely, z(n) := minfk ≥ 1 : n j Fkg). The arithmetic function z : Z≥1 ! Z≥1 is well defined (as can be seen in Lucas [1], p. 300) and, in fact, z(n) ≤ 2n is the sharpest upper bound (as can be seen in [2]). A few values of z(n) (for n 2 [1, 50]) can be found in Table1 (see the Citation: Trojovský, P. On the Parity OEIS [3] sequence A001177 and for more facts on properties of z(n) see, e.g., [4–9]).
    [Show full text]
  • Numbers 1 to 100
    Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number)
    [Show full text]
  • Dan Ismailescu Department of Mathematics, Hofstra University, Hempstead, New York [email protected]
    #A65 INTEGERS 14 (2014) ON NUMBERS THAT CANNOT BE EXPRESSED AS A PLUS-MINUS WEIGHTED SUM OF A FIBONACCI NUMBER AND A PRIME Dan Ismailescu Department of Mathematics, Hofstra University, Hempstead, New York [email protected] Peter C. Shim The Pingry School, 131 Martinsville Road, Basking Ridge, New Jersey [email protected] Received: 12/1/13, Revised: 7/31/14, Accepted: 11/19/14, Published: 12/8/14 Abstract In 2012, Lenny Jones proved that there are infinitely many positive integers that th cannot be expressed as Fn p or Fn + p, where Fn is the n term of the Fibonacci sequence and p denotes a prime.− The smallest integer with this property he could find has 950 digits. We prove that the numbers 135225 and 208641 have this property as well. We also show that there are infinitely many integers that cannot be written as Fn p, for any choice of the signs. In particular, 64369395 is such a number. We answ± ±er a question of Jones by showing that there exist infinitely many integers k such that neither k nor k + 1 can be written as Fn p. Finally, we prove that there exist infinitely many perfect squares and infinitely± many perfect cubes that cannot be written as F p. n ± 1. Introduction In 1849, de Polignac [3] conjectured that every odd integer k can be written as the sum of a prime number and a power of 2. It is well known that de Polignac’s conjecture is false. Erd˝os [1] proved that there are infinitely many odd numbers that cannot be written as 2n p where n is a positive integer and p is a prime.
    [Show full text]
  • Properties of a Generalized Arnold's Discrete Cat
    Master’s thesis Properties of a generalized Arnold’s discrete cat map Author: Fredrik Svanström Supervisor: Per-Anders Svensson Examiner: Andrei Khrennikov Date: 2014-06-06 Subject: Mathematics Level: Second level Course code: 4MA11E Department of Mathematics Abstract After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold’s discrete cat map, including its gen- eralizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell’s cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell’s cat map. From numerical results four conjectures regarding properties of Pell’s cat map are also stated. A brief exposition of some applications of Arnold’s discrete cat map is found in the last part of the thesis. Keywords: Arnold’s discrete cat map, Hyperbolic toral automorphism, Discrete-time dynamical systems, Poincaré recurrence theorem, Number theory, Linear algebra, Fibonacci numbers, Pell numbers, Cryptography. Contents 1 Introduction 1 2 Arnold’s cat map 2 3 The connection between Arnold’s cat and Fibonacci’s rabbits 5 4 Properties of Arnold’s cat map 6 4.1 Periodlengths............................. 6 4.2 Disjointorbits............................. 8 4.3 Higher dimensions of the cat map . 10 4.4 The generalized cat maps with positive unit determinant ..... 11 4.5 Miniatures and ghosts . 12 5 Pell’s cat map 14 5.1 Defining Pell’s cat map . 15 5.2 Some results from elementary number theory .
    [Show full text]
  • Some Open Problems
    Some Open Problems 1. A sequence of Mersenne numbers can be generated by the recursive function ak+1 := 2ak + 1, for all k 2 N, with a0 = 0. We obtain the sequence 1; 3; 7; 15; 31; 63; 127;:::: Note that each term here is of the form 2k − 1, for k 2 N. Any prime in this sequence is called a Mersenne prime, i.e., a prime having the form 2k − 1 for some k 2 N. Prove or disprove that there are infinitely many Mersenne primes! Observations: Note that 15 = 24 − 1 and 63 = 26 − 1 are not Mersenne primes. If k is composite then 2k − 1 is also composite (not prime). If k is composite then k = `m. Then 2k − 1 = (2`)m − 1 = (2` − 1)((2`)m−1 + (2`)m−2 + ::: + 2` + 1): Thus, 2` − 1 divides 2k − 1. If 2k − 1 is a prime then either 2` − 1 = 2k − 1 or 2` − 1 = 1. This implies that k = ` (m = 1) or ` = 1 contradicting the fact that k is composite. Therefore, a necessary condition for a Mersenne number 2k − 1 to be prime is k should be prime. This is not a sufficient condition, for instance, 11 is prime but 211 − 1 = 2047 = 23 × 89 is not a prime. More generally, if ak − 1 is prime then either k = 1 or a = 2. Since, by geometric sum, ak − 1 = (a − 1)(ak−1 + ak−2 + ::: + a + 1) we get ak − 1 is divisible by a − 1. Since ak − 1 is prime either a − 1 = ak − 1 or a − 1 = 1.
    [Show full text]
  • Generalized Pell-Padovan Numbers
    Asian Journal of Advanced Research and Reports 11(2): 8-28, 2020; Article no.AJARR.57839 ISSN: 2582-3248 Generalized Pell-Padovan Numbers ∗ Y ¨ukselSoykan1 1Department of Mathematics, Faculty of Art and Science, Zonguldak Bulent¨ Ecevit University, 67100, Zonguldak, Turkey. Author’s contribution The sole author designed, analyzed, interpreted and prepared the manuscript. Article Information DOI: 10.9734/AJARR/2020/v11i230259 Editor(s): (1) Dr. Hasan Aydogan , Selcuk University, Turkey. Reviewers: (1) B. Selvaraj, Nehru Institute of Engineering and Technology, India. (2) Md. Asaduzzaman, Islamic University, Bangladesh. Complete Peer review History: http://www.sdiarticle4.com/review-history/57839 Received 05 April 2020 Accepted 12 June 2020 Original Research Article Published 20 June 2020 ABSTRACT In this paper, we investigate the generalized Pell-Padovan sequences and we deal with, in detail, four special cases, namely, Pell-Padovan, Pell-Perrin, third order Fibonacci-Pell and third order Lucas-Pell sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences. Keywords: Pell-Padovan numbers; Pell-Perrin numbers; third order Fibonacci-Pell numbers; third order Lucas-Pell numbers. 2010 Mathematics Subject Classification: 11B39, 11B83. 1 INTRODUCTION and third order Lucas-Pell. Before, we recall the generalized Tribonacci sequence and its some The aim of this paper is to define and to explore properties.
    [Show full text]