DOCSLIB.ORG

On Repdigits As Sums of Fibonacci and Tribonacci Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Repdigits As Sums of Fibonacci and Tribonacci Numbers S S symmetry Article On Repdigits as Sums of Fibonacci and Tribonacci Numbers Pavel Trojovský Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic; [email protected]; Tel.: +42-049-333-2860 Received: 17 September 2020; Accepted: 21 October 2020; Published: 26 October 2020 Abstract: In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a “symmetrical” type of numbers) that can be written in the form Fn + Tn, for some n ≥ 1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively. Keywords: Diophantine equations; repdigits; Fibonacci; Tribonacci; Baker’s theory MSC: 11B39; 11J86 1. Introduction A palindromic number is a number that has the same form when written forwards or backwards, i.e., of the form c1c2c3 ... c3c2c1 (thus it can be said that they are “symmetrical” with respect to an axis of symmetry). The first 19th palindromic numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99 and clearly they are a repdigits type. A number n is called repdigit if it has only one repeated digit in its decimal expansion. More precisely, n has the form ! 10` − 1 n = a , 9 for some ` ≥ 1 and a 2 [1, 9] (as usual, we set [a, b] = fa, a + 1, ... , bg, for integers a < b). An old open problem consists in proving the existence of infinitely many prime repunit numbers (sequence A002275 in OEIS [1]), where the `th repunit is defined as 10` − 1 R` = , 9 (it is an easy exercise that if R` is prime, then so is `). There are many articles that address Diophantine equations concerning Fibonacci and Lucas numbers (see, e.g., [2–13]). In the last years, many authors have worked on Diophantine problems related to repdigits (e.g., their sums, concatenations) and linear recurrences (e.g., their product, sums). For more about this subject, we refer the reader to [14–24] and references therein. Remark 1. We remark that the definition of repdigit is not restricted to decimal expansion. In fact, a repdigit in base g ≥ 2, has the form Symmetry 2020, 12, 1774; doi:10.3390/sym12111774 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 1774 2 of 7 ! g` − 1 a , g − 1 for some ` ≥ 1 and a 2 [1, g − 1]. In this work, we shall study two well-known recurrence sequences. The first one is the omnipresent sequence (Fn)n. These numbers are defined by the second order linear recurrence Fn+2 = Fn+1 + Fn, for all n ≥ 0 with initial values F0 = 0 and F1 = 1 (see, e.g., [25]). The sequence of Tribonacci numbers (Tn)n (generalizes the Fibonacci sequence) is defined by the third-order recurrence Tn+3 = Tn+2 + Tn+1 + Tn, for all n ≥ 0 which begins with T0 = 0 and T1 = T2 = 1 (see, e.g., [26,27]). We remark that Luca showed that F10 = 55 is the largest repdigit in the Fibonacci sequence, while Marques [28] proved that the largest repdigit in the Tribonacci sequence is T8 = 44. In this paper, we continue this program by searching for repdigits, which are the sum of a Fibonacci and a Tribonacci number (both with the same index). More specifically, our main result is the following: Theorem 1. The only solutions of the Diophantine equation ! 10` − 1 F + T = a , (1) n n 9 in positive integers (n, `, a), with a 2 [1, 9], are (n, `, a) 2 f(1, 1, 2), (2, 1, 2), (3, 1, 4), (4, 1, 7)g. 2. Auxiliary Results First, we recall a very useful non-recursive formula for the nth Fibonacci numbers. The Binet’s formula is: fn − (−f)−n Fn = p , (2) 5 p where f = (1 + 5)/2. By a simple inductive argument, we can obtain that: n−2 n−1 f ≤ Fn ≤ f , for all n ≥ 1. (3) Also, we can write fn Fn = p + n, (4) 5 p p n where jnj < 1/ 5 (actually, one has the asymptotic formula Fn = (f / 5)(1 + o(1))). For the Tribonacci sequence, in 1982, Spickerman [29] found the following “Binet-like” formula: an bn gn Tn = + + , for all n ≥ 1, (5) −a2 + 4a − 1 −b2 + 4b − 1 −g2 + 4g − 1 p 3 2 1/3 where a, b, g arep the roots of polynomial x − x − x − 1. Numerically, if w1 := (19 + 3 33) and 1/3 w2 := (19 − 3 33) , then Symmetry 2020, 12, 1774 3 of 7 p 1 1 a = 3 (1 + w1 + w2), b = 6 (2 − (w1 + w2) + i 3(w1 − w2)), and g = b. Another very helpful formula provided by Spickermann is the following: a T = an , n (a − b)(a − g) where, as usual, bxe is the nearest integer to x. In particular, it holds that 0 n Tn = a a + h, (6) where jhj < 1/2 and a0 := a/(a − b)(a − g). Moreover, again by an inductive argument, we can deduce that n−2 n−1 a ≤ Tn ≤ a , for all n ≥ 1. (7) The main approach to attack Theorem1 is the Baker’s theory about lower bounds for linear forms in logarithms. The next result is due to Matveev [30] according to Bugeaud, Mignotte and Siksek [9]: Lemma 1. Let a1, a2, a3 2 R be algebraic numbers and let b1, b2, b3 be nonzero integer numbers. Define L = b1 log a1 + b2 log a2 + b3 log a3. Let D = [Q(a1, a2, a3) : Q] (degree of field extension) and let A1, A2, A3 be real numbers such that Aj ≥ maxfDh(aj), j log ajj, 0.16g, for j 2 f1, 2, 3g. Take B ≥ maxf1, maxfjbjjAj/A1; 1 ≤ j ≤ 3gg. If L 6= 0, then 2 log jLj ≥ −C1D A1 A2 A3 log(1.5eDB log(eD)), where 4 5.5 2 C1 = 6750000 · e (20.2 + log(3 D log(eD))). In the previous statement, the logarithmic height of a t-degree algebraic number a is defined by ! 1 t h(a) = log jaj + ∑ log maxf1, ja(j)jg , t j=1 (j) where a is the leading coefficient of the minimal polynomial of a (over Z), (a )1≤j≤t are the algebraic conjugates of a. Some helpful properties of h(x) are in the following lemma (see Property 3.3 of [31]): Lemma 2. Let x and y be algebraic numbers. Then h(xy) ≤ h(x) + h(y); h(x + y) ≤ h(x) + h(y) + log 2; h(ar) = jrj · h(a), for all r 2 Q. Our last ingredient is a reduction method provided by Dujella and Peth˝o[32], which is itself a variation of the result of Baker and Davenport [33]. For x 2 R, set kxk = minfjx − nj : n 2 Zg = jx − bxej for the distance from x to the nearest integer. Symmetry 2020, 12, 1774 4 of 7 Lemma 3. For a positive integer M, let p/q be a convergent of the continued fraction of g 62 Q, such that q > 6M, and let m, A and B be real numbers, with A > 0 and B > 1. If the number e = kmqk − Mkgqk is positive, then there is no solution to the Diophantine inequality 0 < mg − n + m < A · B−m in integers m, n > 0 with log(Aq/e) ≤ m < M. log B See Lemma 5 of [32]. Now, we are in a position to prove our main theorem. 3. The Proof of Theorem1 3.1. Finding an Upper Bound for n and ` By using (4) and (6) in (1), we have ! fn 10` − 1 p + n + a0an + h = a . 5 9 We can rewrite the previous equality as ` 0 n 10 n a a − a < 2.4f , (8) 9 p where we used that jnj ≤ 1/ 5 and jhj < 1/2. After dividing by a0an, we obtain a 8 1 − a−n10` < , (9) 9a0 (1.13)n where we used that a/f > 1.13 and a0 > 0.3. Let us define L = ` log 10 − n log a + log qa, 0 where qa := a/9a , with a 2 [1, 9]. It follows from (9) that 8 jeL − 1j < . (10) (1.13)n ` n Now, we claim that L is nonzero. Indeed, on the contrary, we would have 10 qa = a and so a · 10` = 9a0an. However, the minimal polynomial of a0, namely 44x3 − 2x − 1, has all its roots inside the unit circle and also jbj = jgj < 1. Thus, we can conjugate the relation a · 10` = 9a0an by the Galois automorphism a ! b in order to obtain a · 10` = 9(b0)bn. By applying absolute values in the previous expression, we get 10` ≤ jaj · 10` = 9jb0jjbjn < 9 which contradicts the fact that ` ≥ 1. When L > 0, then L < eL − 1 < 8 · (1.13)−n, while for L < 0, we can use that 1 − e−jLj = jeL − 1j < 8 · (1.13)−n to infer that 8 · (1.13)−n jLj < ejLj − 1 < < 8 · (1.13)−n+1. 1 − 8 · (1.13)−n Symmetry 2020, 12, 1774 5 of 7 Hence, we have jLj < 8 × (1.13)−n+1. Therefore log jLj < −(n − 1) log(1.13) + log 8. (11) Now, we can apply Lemma1 for the choice of a1 := 10, a2 := a, a3 = qa, b1 = `, b2 = −n, b3 = 1, 0 0 10 where qa := a/(9a ).
Load more

Recommended publications
  • An Amazing Prime Heuristic.Pdf
    This document has been moved to https://arxiv.org/abs/2103.04483 Please use that version instead. AN AMAZING PRIME HEURISTIC CHRIS K. CALDWELL 1. Introduction The record for the largest known twin prime is constantly changing. For example, in October of 2000, David Underbakke found the record primes: 83475759 264955 1: · The very next day Giovanni La Barbera found the new record primes: 1693965 266443 1: · The fact that the size of these records are close is no coincidence! Before we seek a record like this, we usually try to estimate how long the search might take, and use this information to determine our search parameters. To do this we need to know how common twin primes are. It has been conjectured that the number of twin primes less than or equal to N is asymptotic to N dx 2C2N 2C2 2 2 Z2 (log x) ∼ (log N) where C2, called the twin prime constant, is approximately 0:6601618. Using this we can estimate how many numbers we will need to try before we find a prime. In the case of Underbakke and La Barbera, they were both using the same sieving software (NewPGen1 by Paul Jobling) and the same primality proving software (Proth.exe2 by Yves Gallot) on similar hardware{so of course they choose similar ranges to search. But where does this conjecture come from? In this chapter we will discuss a general method to form conjectures similar to the twin prime conjecture above. We will then apply it to a number of different forms of primes such as Sophie Germain primes, primes in arithmetic progressions, primorial primes and even the Goldbach conjecture.
    [Show full text]
  • Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
    TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube.
    [Show full text]
  • Twelve Simple Algorithms to Compute Fibonacci Numbers Arxiv
    Twelve Simple Algorithms to Compute Fibonacci Numbers Ali Dasdan KD Consulting Saratoga, CA, USA [email protected] April 16, 2018 Abstract The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used in introductory algorithms courses. In this paper, we present twelve of these well-known algo- rithms and some of their properties. These algorithms, though very simple, illustrate multiple concepts from the algorithms field, so we highlight them. We also present the results of a small-scale experi- mental comparison of their runtimes on a personal laptop. Finally, we provide a list of homework questions for the students. We hope that this paper can serve as a useful resource for the students learning the basics of algorithms. arXiv:1803.07199v2 [cs.DS] 13 Apr 2018 1 Introduction The Fibonacci numbers are a sequence Fn of integers in which every num- ber after the first two, 0 and 1, is the sum of the two preceding num- bers: 0; 1; 1; 2; 3; 5; 8; 13; 21; ::. More formally, they are defined by the re- currence relation Fn = Fn−1 + Fn−2, n ≥ 2 with the base values F0 = 0 and F1 = 1 [1, 5, 7, 8]. 1 The formal definition of this sequence directly maps to an algorithm to compute the nth Fibonacci number Fn. However, there are many other ways of computing the nth Fibonacci number.
    [Show full text]
  • Number World Number Game "Do You Like Number Games?", Zaina Teacher Asked
    1 Number World Number game "Do you like number games?", Zaina teacher asked. "Oh! Yes!", said the children. "I'll say a number; you give me the next number at once. Ready?" "Ready!" "Ten", teacher began. "Eleven", said all the children. "Forty three" "Forty four" The game went on. "Four thousand ninety nine", teacher said. "Five thousand", replied some one. "Oh! No!... Four thousand and hundred", some caught on. Such mistakes are common. Try this on your friends. First Day Fiesta What is the number of children in First Day Fiesta class 1? What is the largest number you can read? What is the largest four-digit number? What is the next number? 435268 children in Class 1. 8 And the largest five-digit number? What is the next number? How do we find this number? Giant number How do we read it? Look at the table of large numbers: If we are asked for a large number, we often say crore or hundred crore. Put- 1 One ting ten zeros after one makes thou- 10 Ten sand crore. Think about the size of the number with hundred zeros after one. 100 Hundred This is called googol. This name was 1000 Thousand popularized by Edward Kasner in 10000 Ten thousand 1938. 100000 Lakh In most countries, one lakh is named hundred thousand and ten lakh is 1000000 Ten lakh named million. 10000000 Crore 100000000 Ten crore You're always counting This continues with hundred crore, thousand numbers! What's crore, and so on. your goal? Now can you say what we get when we add one Googol! to ninety nine thousand nine hundred and ninety nine? 99999 + 1 = 100000 How do we read this? Look it up in the table.
    [Show full text]
  • Input for Carnival of Math: Number 115, October 2014
    Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144.
    [Show full text]
  • Crazy Representations of Natural Numbers, Selfie Numbers
    Inder J. Taneja RGMIA Research Report Collection, 19(2016), pp.1-37, http://rgmia.org/v19.php Crazy Representations of Natural Numbers, Selfie Numbers, Fibonacci Sequence, and Selfie Fractions Inder J. Taneja1 SUMMARY This summary brings author’s work on numbers. The study is made in different ways. Specially, towards, Crazy Representations of Natural Numbers, Running Expressions, Selfie Numbers etc. Natural numbers are represented in different situations, such as, writing in terms of 1 to 9 or reverse, flexible power of same digits as bases, single digit, single letter, etc. Expressions appearing with equalities having 1 to 9 or 9 to 1 or 9 to 0, calling running expressions are also presented. In continuation, there is work is on selfie numbers, unified, patterns, symmetrical representations in selfie numbers, Fibonacci sequence and selfie numbers, flexible power Narcissistic and selfie numbers, selfie fractions, etc. The selfie numbers may also be considered as generalized or wild narcissistic numbers, where natural numbers are represented by their own digits with certain operations. The study is also made towards equivalent fractions and palindromic-type numbers. This summary is separated by sections and subsections given as follows: 1 Crazy Representations of Natural Numbers [1]; 2 Flexible Power Representations [34]; 2.1 Unequal String Lengths [33]; 2.2 Equal String Lengths [25]; 3 Pyramidical Representations [20, 24, 31, 32]; 3.1 Crazy Representations [33]; 3.2 Flexible Power [25]; 4 Double Sequential Representations [20, 24, 31, 32]; 5 Triple Sequential Representations; [35]; 6 Single Digit Representations; [2]; 7 Single Letter Representations [4, 8]; 7.1 Single Letter Power Representations [8]; 7.2 Palindromic and Number Patterns [9, 10]; 8 Running Expressions.
    [Show full text]
  • Fibonacci Numbers
    mathematics Article On (k, p)-Fibonacci Numbers Natalia Bednarz The Faculty of Mathematics and Applied Physics, Rzeszow University of Technology, al. Powsta´nców Warszawy 12, 35-959 Rzeszów, Poland; [email protected] Abstract: In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultane- ously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too. Keywords: Fibonacci numbers; Pell numbers; Narayana numbers MSC: 11B39; 11B83; 11C20 1. Introduction By numbers of the Fibonacci type we mean numbers defined recursively by the r-th order linear recurrence relation of the form an = b1an−1 + b2an−2 + ··· + bran−r, for n > r, (1) where r > 2 and bi > 0, i = 1, 2, ··· , r are integers. For special values of r and bi, i = 1, 2, ··· r, the Equality (1) defines well-known numbers of the Fibonacci type and their generalizations. We list some of them: Citation: Bednarz, N. On 1. Fibonacci numbers: Fn = Fn−1 + Fn−2 for n > 2, with F0 = F1 = 1. (k, p)-Fibonacci Numbers. 2. Lucas numbers: Ln = Ln−1 + Ln−2 for n > 2, with L0 = 2, L1 = 1. Mathematics 2021, 9, 727. https:// 3. Pell numbers: Pn = 2Pn−1 + Pn−2 for n > 2, with P0 = 0, P1 = 1. doi.org/10.3390/math9070727 4. Pell–Lucas numbers: Qn = 2Qn−1 + Qn−2 for n > 2, with Q0 = 1, Q1 = 3. 5. Jacobsthal numbers: Jn = Jn−1 + 2Jn−2 for n > 2, with J0 = 0, J1 = 1.
    [Show full text]
  • On Repdigits As Product of Consecutive Fibonacci Numbers1
    Rend. Istit. Mat. Univ. Trieste Volume 44 (2012), 393–397 On repdigits as product of consecutive Fibonacci numbers1 Diego Marques and Alain Togbe´ Abstract. Let (Fn)n≥0 be the Fibonacci sequence. In 2000, F. Luca proved that F10 = 55 is the largest repdigit (i.e. a number with only one distinct digit in its decimal expansion) in the Fibonacci sequence. In this note, we show that if Fn ··· Fn+(k−1) is a repdigit, with at least two digits, then (k, n) = (1, 10). Keywords: Fibonacci, repdigits, sequences (mod m) MS Classification 2010: 11A63, 11B39, 11B50 1. Introduction Let (Fn)n≥0 be the Fibonacci sequence given by Fn+2 = Fn+1 + Fn, for n ≥ 0, where F0 = 0 and F1 = 1. These numbers are well-known for possessing amaz- ing properties. In 1963, the Fibonacci Association was created to provide an opportunity to share ideas about these intriguing numbers and their applica- tions. We remark that, in 2003, Bugeaud et al. [2] proved that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 (see [6] for the Fibono- mial version). In 2005, Luca and Shorey [5] showed, among other things, that a non-zero product of two or more consecutive Fibonacci numbers is never a perfect power except for the trivial case F1 · F2 = 1. Recall that a positive integer is called a repdigit if it has only one distinct digit in its decimal expansion. In particular, such a number has the form a(10m − 1)/9, for some m ≥ 1 and 1 ≤ a ≤ 9.
    [Show full text]
  • Product of Consecutive Tribonacci Numbers with Only One Distinct Digit
    1 2 Journal of Integer Sequences, Vol. 22 (2019), 3 Article 19.6.3 47 6 23 11 Product of Consecutive Tribonacci Numbers With Only One Distinct Digit Eric F. Bravo and Carlos A. G´omez Department of Mathematics Universidad del Valle Calle 13 No 100 – 00 Cali Colombia [email protected] [email protected] Florian Luca School of Mathematics University of the Witwatersrand Johannesburg South Africa and Research Group in Algebraic Structures and Applications King Abdulaziz University Jeddah Saudi Arabia and Department of Mathematics University of Ostrava 30 Dubna 22, 701 03 Ostrava 1 Czech Republic [email protected] Abstract Let (Fn)n≥0 be the sequence of Fibonacci numbers. Marques and Togb´eproved that if the product Fn ··· Fn+ℓ−1 is a repdigit (i.e., a number with only distinct digit 1 in its decimal expansion), with at least two digits, then (ℓ,n)=(1, 10). In this paper, we solve the same problem with Tribonacci numbers instead of Fibonacci numbers. 1 Introduction A positive integer is called a repdigit if it has only one distinct digit in its decimal expansion. The sequence of numbers with repeated digits is included in Sloane’s On-Line Encyclopedia of Integer Sequences (OEIS) [11] as sequence A010785. In 2000, Luca [6] showed that the largest repdigit Fibonacci number is F10 = 55 and the largest repdigit Lucas number is L5 = 11. Motivated by the results of Luca [6], several authors have explored repdigits in general- izations of Fibonacci numbers and Lucas numbers. For instance, Bravo and Luca [1] showed 1 (3) that the only repdigit in the k-generalized Fibonacci sequence , is F8 = 44.
    [Show full text]
  • Cullen Numbers with the Lehmer Property
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]).
    [Show full text]
  • Retrograde Renegades and the Pascal Connection: Repeating Decimals Represented by Fibonacci and Other Sequences Appearing from Right to Left
    RETROGRADE RENEGADES AND THE PASCAL CONNECTION: REPEATING DECIMALS REPRESENTED BY FIBONACCI AND OTHER SEQUENCES APPEARING FROM RIGHT TO LEFT Marjorie Bicknell-Johnson Santa Clara Unified School District, Santa Clara, CA 95051 (Submitted October 1987) Repeating decimals show a surprisingly rich variety of number sequence patterns when their repetends are viewed in retrograde fashion, reading from the rightmost digit of the repeating cycle towards the left. They contain geometric sequences as well as Fibonacci numbers generated by an application of Pascal*s triangle. Further, fractions whose repetends end with successive terms of Fnm , m = 1, 2, ..., occurring in repeating blocks of k digits, are completely characterized, as well as fractions ending with Fnm+p or Lnm+p, th where Fn is the n Fibonacci number, F1 = 1, F2 = 1, Fn+i = Fn + Fn_19 th and Ln is the n Lucas number, 1. The Pascal Connection It is no surprise that 1/89 contains the sum of successive Fibonacci num- bers in its decimal expansion [2], [3], [4], [5], as 1/89 = .012358 13 21 34 However, 1/89 can also be expressed as the sum of successive powers of 11, as 1/89 = .01 .0011 .000121 .00001331 .0000014641 where 1/89 = 1/102 + ll/lO4 + 1'12/106 + ..., which is easily shown by summing the geometric progression. If the array above had the leading zeros removed and was left-justified, we would have Pascal's triangle in a form where the Fibonacci numbers arise as the sum of the rising diagonals. Notice that llk generates rows of Pascal's triangle, and that the columns of the array expressing 1/89 are the diagonals of Pascal's triangle.
    [Show full text]
  • Enciclopedia Matematica a Claselor De Numere Întregi
    THE MATH ENCYCLOPEDIA OF SMARANDACHE TYPE NOTIONS vol. I. NUMBER THEORY Marius Coman INTRODUCTION About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name), it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. In this first volume of encyclopedia we try to synthesize his work in the field of number theory, one of the great Smarandache’s passions, a surfer on the ocean of numbers, to paraphrase the title of the book Surfing on the ocean of numbers – a few Smarandache notions and similar topics, by Henry Ibstedt. We quote from the introduction to the Smarandache’work “On new functions in number theory”, Moldova State University, Kishinev, 1999: “The performances in current mathematics, as the future discoveries, have, of course, their beginning in the oldest and the closest of philosophy branch of nathematics, the number theory. Mathematicians of all times have been, they still are, and they will be drawn to the beaty and variety of specific problems of this branch of mathematics. Queen of mathematics, which is the queen of sciences, as Gauss said, the number theory is shining with its light and attractions, fascinating and facilitating for us the knowledge of the laws that govern the macrocosm and the microcosm”.
    [Show full text]