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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. -
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. -
Second Grade Unit 1: Extending Base Ten Understanding
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Second Grade Unit 1: Extending Base Ten Understanding These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department of Education Georgia Standards of Excellence Framework GSE Extending Base Ten Understanding Unit 1 Unit 1: Extending Base Ten Understanding TABLE OF CONTENTS Overview……………………………………………………………………………..... 3 Standards for Mathematical Practice …………………………………………………. 5 Standards for Mathematical Content …………………………………………………. 6 Big Ideas ………………………………………………………………………………. 6 Essential Questions ……………………………………………………………………….. 7 Concepts and Skills to Maintain ………………………………………………………. 7 Strategies for Teaching and Learning ……………………………………………. 8 Selected Terms and Symbols ………………………………………………..………… 10 Task Types ………………………………………………………………...………….. 12 Task Descriptions …………………………………………………………………….. 13 Intervention Table………………….………………………………………………….. 15 Straws! Straws! Straws!.........................................................................… 16 Where Am I on the Number Line? …………………………………… 23 I Spy a Number ………………………………………………………. 31 Number Hop …………………………………………………………… 38 Place Value Play …………………………………………………….... 44 The Importance of Zero ………………………………………………. 51 Base Ten Pictures …………………………………………………….. 62 Building Base Ten Numbers …………………………………………. 69 What's My Number? …………………………………………………. 74 Capture the Caterpillar ……………………………………………….. 79 Formative Assessment Lesson ……………………………………….. 88 Fill the Bucket ……………………………………………………….. -
Under the Direction of Denise S
EXPLORING MIDDLE GRADE TEACHERS’ KNOWLEDGE OF PARTITIVE AND QUOTITIVE FRACTION DIVISION by SOO JIN LEE (Under the Direction of Denise S. Mewborn) ABSTRACT The purpose of the present qualitative study was to investigate middle grades (Grade 5-7) mathematics teachers’ knowledge of partitive and quotitive fraction division. Existing research has documented extensively that preservice and inservice teachers lack adequate preparation in the mathematics they teach (e.g., Ball, 1990, 1993). Especially, research on teachers’ understanding of fraction division (e.g., Ball, 1990; Borko, 1992; Ma, 1999; Simon, 1993) has demonstrated that one or more pieces of an ideal knowledge package (Ma, 1999) for fraction division is missing. Although previous studies (e.g., Ball, 1990; Borko, 1992; Ma, 1999; Simon, 1993; Tirosh & Graeber, 1989) have stressed errors and constraints on teachers’ knowledge of fraction division, few studies have been conducted to explore teachers’ knowledge of fraction division at a fine-grained level (Izsák, 2008). Thus, I concentrated on teachers’ operations and flexibilities with conceptual units in partitive and quotitive fraction division situations. Specifically, I attempted to develop a model of teachers’ ways of knowing fraction division by observing their performance through a sequence of division problems in which the mathematical relationship between the dividend and the divisor became increasingly complex. This is my first step toward building a learning trajectory of teachers’ ways of thinking, which can be extremely useful for thinking about how to build an effective professional development program and a teacher education program. The theoretical frame that I developed for this study emerged through analyses of teachers’ participation in the professional development program where they were encouraged to reason with/attend to quantitative units using various drawings such as length and area models. -
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. -
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. -
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 . -
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 = − . -
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 -
Counting Keith Numbers
1 2 Journal of Integer Sequences, Vol. 10 (2007), 3 Article 07.2.2 47 6 23 11 Counting Keith Numbers Martin Klazar Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI) Faculty of Mathematics and Physics Charles University Malostransk´en´am. 25 11800 Praha Czech Republic [email protected] Florian Luca Instituto de Matem´aticas Universidad Nacional Autonoma de M´exico C.P. 58089 Morelia, Michoac´an M´exico [email protected] Abstract A Keith number is a positive integer N with the decimal representation a a a 1 2 ··· n such that n 2 and N appears in the sequence (K ) given by the recurrence ≥ m m≥1 K = a ,...,K = a and K = K + K + + K for m>n. We prove 1 1 n n m m−1 m−2 ··· m−n that there are only finitely many Keith numbers using only one decimal digit (i.e., a = a = = a ), and that the set of Keith numbers is of asymptotic density zero. 1 2 ··· n 1 Introduction With the number 197, let (Km)m≥1 be the sequence whose first three terms K1 =1, K2 =9 and K3 = 7 are the digits of 197 and that satisfies the recurrence Km = Km−1 +Km−2 +Km−3 1 for all m> 3. Its initial terms are 1, 9, 7, 17, 33, 57, 107, 197, 361, 665,... Note that 197 itself is a member of this sequence. This phenomenon was first noticed by Mike Keith and such numbers are now called Keith numbers. More precisely, a number N with decimal representation a1a2 an is a Keith number if n 2 and N appears in the N N ··· ≥ sequence K = (Km )m≥1 whose n initial terms are the digits of N read from left to right N N N N and satisfying Km = Km−1 + Km−2 + + Km−n for all m>n. -
Observations on Icosagonal Pyramidal Number
International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 7 (July 2013), PP. 32-37 www.irjes.com Observations On Icosagonal Pyramidal Number 1M.A.Gopalan, 2Manju Somanath, 3K.Geetha, 1Department of Mathematics, Shrimati Indira Gandhi college, Tirchirapalli- 620 002. 2Department of Mathematics, National College, Trichirapalli-620 001. 3Department of Mathematics, Cauvery College for Women, Trichirapalli-620 018. ABSTRACT:- We obtain different relations among Icosagonal Pyramidal number and other two, three and four dimensional figurate numbers. Keyword:- Polygonal number, Pyramidal number, Centered polygonal number, Centered pyramidal number, Special number MSC classification code: 11D99 I. INTRODUCTION Fascinated by beautiful and intriguing number patterns, famous mathematicians, share their insights and discoveries with each other and with readers. Throughout history, number and numbers [2,3,7-15] have had a tremendous influence on our culture and on our language. Polygonal numbers can be illustrated by polygonal designs on a plane. The polygonal number series can be summed to form “solid” three dimensional figurate numbers called Pyramidal numbers [1,4,5 and 6] that can be illustrated by pyramids. In this communication we 32 20 65n n n deal with Icosagonal Pyramidal numbers given by p and various interesting relations n 2 among these numbers are exhibited by means of theorems involving the relations. Notation m pn = Pyramidal number of rank n with sides m tmn, = Polygonal number of rank n with sides m jaln = Jacobsthal Lucas number ctmn, = Centered Polygonal number of rank n with sides m m cpn = Centered Pyramidal number of rank n with sides m gn = Gnomonic number of rank n with sides m pn = Pronic number carln = Carol number mern = Mersenne number, where n is prime culn = Cullen number Than = Thabit ibn kurrah number II. -
Phd Complete Document
Idiopathy: A Novel and ‘A Failed Entertainment:’ The Process of Paying Attention in and to Infinite Jest by Sam Byers Thesis submitted for the qualification of Doctor of Philosophy at The University of East Anglia, Department of Literature, Drama, and Creative Writing, September 2013. © This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognize that its copyright rests with the author and that use of any information derived there from must be in accordance with current UK Copyright Law. In addition, any quotation or extract must include full attribution. WORD COUNT: 90,000 words + Appendix 2 Abstract My PhD thesis combines a novel, Idiopathy, and a critical essay, ‘A Failed Entertainment:’ The Process of Paying Attention In and To Infinite Jest. Idiopathy examines the inner lives of three friends and their experiences of both separation and a reunion. By emphasising internality over externality it seeks to examine the notion that how we feel about our lives may be more important, indeed, more ‘real,’ than what takes place around us. Idiopathy is overtly concerned with the extent to which interaction with others is problematic, and pays particular attention to the emotional complexities of friendships, relationships and family dynamics. Through these pained and frequently unsatisfactory interactions the novel seeks to satirise broader social phenomena, such as media panics, health epidemics, ideological protest movements and the rise of what might be termed a ‘pathologised’ culture, in which all the characters are continually concerned that something is ‘wrong’ with them. Illness is a metaphor throughout, and is explored not only through the characters’ sense of personal malaise, but also through the device of a fictional cattle epidemic that forms a media backdrop to unfolding events.