Module 1B Chapter 3 Pages 25 – 38
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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. -
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. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
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. -
Sequences of Primes Obtained by the Method of Concatenation
SEQUENCES OF PRIMES OBTAINED BY THE METHOD OF CONCATENATION (COLLECTED PAPERS) Copyright 2016 by Marius Coman Education Publishing 1313 Chesapeake Avenue Columbus, Ohio 43212 USA Tel. (614) 485-0721 Peer-Reviewers: Dr. A. A. Salama, Faculty of Science, Port Said University, Egypt. Said Broumi, Univ. of Hassan II Mohammedia, Casablanca, Morocco. Pabitra Kumar Maji, Math Department, K. N. University, WB, India. S. A. Albolwi, King Abdulaziz Univ., Jeddah, Saudi Arabia. Mohamed Eisa, Dept. of Computer Science, Port Said Univ., Egypt. EAN: 9781599734668 ISBN: 978-1-59973-466-8 1 INTRODUCTION The definition of “concatenation” in mathematics is, according to Wikipedia, “the joining of two numbers by their numerals. That is, the concatenation of 69 and 420 is 69420”. Though the method of concatenation is widely considered as a part of so called “recreational mathematics”, in fact this method can often lead to very “serious” results, and even more than that, to really amazing results. This is the purpose of this book: to show that this method, unfairly neglected, can be a powerful tool in number theory. In particular, as revealed by the title, I used the method of concatenation in this book to obtain possible infinite sequences of primes. Part One of this book, “Primes in Smarandache concatenated sequences and Smarandache-Coman sequences”, contains 12 papers on various sequences of primes that are distinguished among the terms of the well known Smarandache concatenated sequences (as, for instance, the prime terms in Smarandache concatenated odd -
Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected]
Transformations Volume 2 Article 5 Issue 2 Winter 2016 12-27-2016 Number Pattern Hui Fang Huang Su Nova Southeastern University, [email protected] Denise Gates Janice Haramis Farrah Bell Claude Manigat See next page for additional authors Follow this and additional works at: https://nsuworks.nova.edu/transformations Part of the Curriculum and Instruction Commons, Science and Mathematics Education Commons, Special Education and Teaching Commons, and the Teacher Education and Professional Development Commons Recommended Citation Su, Hui Fang Huang; Gates, Denise; Haramis, Janice; Bell, Farrah; Manigat, Claude; Hierpe, Kristin; and Da Silva, Lourivaldo (2016) "Number Pattern," Transformations: Vol. 2 : Iss. 2 , Article 5. Available at: https://nsuworks.nova.edu/transformations/vol2/iss2/5 This Article is brought to you for free and open access by the Abraham S. Fischler College of Education at NSUWorks. It has been accepted for inclusion in Transformations by an authorized editor of NSUWorks. For more information, please contact [email protected]. Number Pattern Cover Page Footnote This article is the result of the MAT students' collaborative research work in the Pre-Algebra course. The research was under the direction of their professor, Dr. Hui Fang Su. The ap per was organized by Team Leader Denise Gates. Authors Hui Fang Huang Su, Denise Gates, Janice Haramis, Farrah Bell, Claude Manigat, Kristin Hierpe, and Lourivaldo Da Silva This article is available in Transformations: https://nsuworks.nova.edu/transformations/vol2/iss2/5 Number Patterns Abstract In this manuscript, we study the purpose of number patterns, a brief history of number patterns, and classroom uses for number patterns and magic squares. -
The Prime Number Theorem a PRIMES Exposition
The Prime Number Theorem A PRIMES Exposition Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao TABLE OF CONTENTS 1 Introduction 2 Tools from Complex Analysis 3 Entire Functions 4 Hadamard Factorization Theorem 5 Riemann Zeta Function 6 Chebyshev Functions 7 Perron Formula 8 Prime Number Theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 2 Introduction • Euclid (300 BC): There are infinitely many primes • Legendre (1808): for primes less than 1,000,000: x π(x) ' log x © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 3 Progress on the Distribution of Prime Numbers • Euler: The product formula 1 X 1 Y 1 ζ(s) := = ns 1 − p−s n=1 p so (heuristically) Y 1 = log 1 1 − p−1 p • Chebyshev (1848-1850): if the ratio of π(x) and x= log x has a limit, it must be 1 • Riemann (1859): On the Number of Primes Less Than a Given Magnitude, related π(x) to the zeros of ζ(s) using complex analysis • Hadamard, de la Vallée Poussin (1896): Proved independently the prime number theorem by showing ζ(s) has no zeros of the form 1 + it, hence the celebrated prime number theorem © Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao 4 Tools from Complex Analysis Theorem (Maximum Principle) Let Ω be a domain, and let f be holomorphic on Ω. (A) jf(z)j cannot attain its maximum inside Ω unless f is constant. (B) The real part of f cannot attain its maximum inside Ω unless f is a constant. Theorem (Jensen’s Inequality) Suppose f is holomorphic on the whole complex plane and f(0) = 1. -
Ramanujan, Robin, Highly Composite Numbers, and the Riemann Hypothesis
Contemporary Mathematics Volume 627, 2014 http://dx.doi.org/10.1090/conm/627/12539 Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis Jean-Louis Nicolas and Jonathan Sondow Abstract. We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises. 1. Mathematical Background Definition. The sum-of-divisors function σ is defined by 1 σ(n):= d = n . d d|n d|n In 1913, Gr¨onwall found the maximal order of σ. Gr¨onwall’s Theorem [8]. The function σ(n) G(n):= (n>1) n log log n satisfies lim sup G(n)=eγ =1.78107 ... , n→∞ where 1 1 γ := lim 1+ + ···+ − log n =0.57721 ... n→∞ 2 n is the Euler-Mascheroni constant. Gr¨onwall’s proof uses: Mertens’s Theorem [10]. If p denotes a prime, then − 1 1 1 lim 1 − = eγ . x→∞ log x p p≤x 2010 Mathematics Subject Classification. Primary 01A60, 11M26, 11A25. Key words and phrases. Riemann Hypothesis, Ramanujan’s Theorem, Robin’s Theorem, sum-of-divisors function, highly composite number, superabundant, colossally abundant, Euler phi function. ©2014 American Mathematical Society 145 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 146 JEAN-LOUIS NICOLAS AND JONATHAN SONDOW Figure 1. Thomas Hakon GRONWALL¨ (1877–1932) Figure 2. Franz MERTENS (1840–1927) Nowwecometo: Ramanujan’s Theorem [2, 15, 16]. -
On Types of Elliptic Pseudoprimes
journal of Groups, Complexity, Cryptology Volume 13, Issue 1, 2021, pp. 1:1–1:33 Submitted Jan. 07, 2019 https://gcc.episciences.org/ Published Feb. 09, 2021 ON TYPES OF ELLIPTIC PSEUDOPRIMES LILJANA BABINKOSTOVA, A. HERNANDEZ-ESPIET,´ AND H. Y. KIM Boise State University e-mail address: [email protected] Rutgers University e-mail address: [email protected] University of Wisconsin-Madison e-mail address: [email protected] Abstract. We generalize Silverman's [31] notions of elliptic pseudoprimes and elliptic Carmichael numbers to analogues of Euler-Jacobi and strong pseudoprimes. We inspect the relationships among Euler elliptic Carmichael numbers, strong elliptic Carmichael numbers, products of anomalous primes and elliptic Korselt numbers of Type I, the former two of which we introduce and the latter two of which were introduced by Mazur [21] and Silverman [31] respectively. In particular, we expand upon the work of Babinkostova et al. [3] on the density of certain elliptic Korselt numbers of Type I which are products of anomalous primes, proving a conjecture stated in [3]. 1. Introduction The problem of efficiently distinguishing the prime numbers from the composite numbers has been a fundamental problem for a long time. One of the first primality tests in modern number theory came from Fermat Little Theorem: if p is a prime number and a is an integer not divisible by p, then ap−1 ≡ 1 (mod p). The original notion of a pseudoprime (sometimes called a Fermat pseudoprime) involves counterexamples to the converse of this theorem. A pseudoprime to the base a is a composite number N such aN−1 ≡ 1 mod N. -
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, .. -
Mathematics at Allerton Thoughts on Facilitating Some Teacher Development
ECARDA Mathematics at Allerton Thoughts on facilitating some teacher development Fifty years after starting my own career in mathematics education, I have been invited to work with a primary school on developing their teaching and learning of mathematics. This pamphlet records my preparation and includes some of the mathematical activities used. Peter Lacey May 2016 Musing over the preparation of some development work with teachers I have been invited to work with teachers on their mathematics and its teaching fifty years after I started my own learning journey as a teaching majoring on mathematics. The state educational landscape has altered beyond recognition since then but the disconnect between mathematics and the majority of the English population remains. Other than the language used, there is little change to the stream of reports bemoaning the state of English mathematics education and the consequent low levels of mathematical competence of school leavers. Is it really that bad? Have things really got worse? A lot of money has been invested in improving the teaching and learning of mathematics, not least on versions of a national curriculum, versions of end-of-key- stage testing and examinations, national strategies and ever more “focussed” school inspections. Has all this been wasted? I still smile at the recollection of a professor of education who told me that everything except the evidence points to a decline in standards of mathematics! I bring to mind the 1984 – 1987 “Raising Achievement Project” (RAMP) which, amongst other things, paid to release teachers of mathematics from their classrooms for one day a week over the three years to reflect on, research and re-engineer their approaches to teaching. -
2 Primes Numbers
V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 2 Primes Numbers Definition 2.1 A number is prime is it is greater than 1, and its only divisors are itself and 1. A number is called composite if it is greater than 1 and is the product of two numbers greater than 1. Thus, the positive numbers are divided into three mutually exclusive classes. The prime numbers, the composite numbers, and the unit 1. We can show that a number is composite numbers by finding a non-trivial factorization.8 Thus, 21 is composite because 21 = 3 7. The letter p is usually used to denote a prime number. × The first fifteen prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 These are found by a process of elimination. Starting with 2, 3, 4, 5, 6, 7, etc., we eliminate the composite numbers 4, 6, 8, 9, and so on. There is a systematic way of finding a table of all primes up to a fixed number. The method is called a sieve method and is called the Sieve of Eratosthenes9. We first illustrate in a simple example by finding all primes from 2 through 25. We start by listing the candidates: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 The first number 2 is a prime, but all multiples of 2 after that are not. Underline these multiples. They are eliminated as composite numbers. The resulting list is as follows: 2, 3, 4,5,6,7,8,9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25 The next number not eliminated is 3, the next prime.