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On Untouchable Numbers and Related Problems
ON UNTOUCHABLE NUMBERS AND RELATED PROBLEMS CARL POMERANCE AND HEE-SUNG YANG Abstract. In 1973, Erd˝os proved that a positive proportion of numbers are untouchable, that is, not of the form σ(n) n, the sum of the proper divisors of n. We investigate the analogous question where σ is− replaced with the sum-of-unitary-divisors function σ∗ (which sums divisors d of n such that (d, n/d) = 1), thus solving a problem of te Riele from 1976. We also describe a fast algorithm for enumerating untouchable numbers and their kin. 1. Introduction If f(n) is an arithmetic function with nonnegative integral values it is interesting to con- sider Vf (x), the number of integers 0 m x for which f(n) = m has a solution. That is, one might consider the distribution≤ of the≤ range of f within the nonnegative integers. For some functions f(n) this is easy, such as the function f(n) = n, where Vf (x) = x , or 2 ⌊ ⌋ f(n) = n , where Vf (x) = √x . For f(n) = ϕ(n), Euler’s ϕ-function, it was proved by ⌊ ⌋ 1+o(1) Erd˝os [Erd35] in 1935 that Vϕ(x) = x/(log x) as x . Actually, the same is true for a number of multiplicative functions f, such as f = σ,→ the ∞ sum-of-divisors function, and f = σ∗, the sum-of-unitary-divisors function, where we say d is a unitary divisor of n if d n, | d > 0, and gcd(d,n/d) = 1. In fact, a more precise estimation of Vf (x) is known in these cases; see Ford [For98]. -
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. -
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. -
On Sufficient Conditions for the Existence of Twin Values in Sieves
On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers by Luke Szramowski Submitted in Partial Fulfillment of the Requirements For the Degree of Masters of Science in the Mathematics Program Youngstown State University May, 2020 On Sufficient Conditions for the Existence of Twin Values in Sieves over the Natural Numbers Luke Szramowski I hereby release this thesis to the public. I understand that this thesis will be made available from the OhioLINK ETD Center and the Maag Library Circulation Desk for public access. I also authorize the University or other individuals to make copies of this thesis as needed for scholarly research. Signature: Luke Szramowski, Student Date Approvals: Dr. Eric Wingler, Thesis Advisor Date Dr. Thomas Wakefield, Committee Member Date Dr. Thomas Madsen, Committee Member Date Dr. Salvador A. Sanders, Dean of Graduate Studies Date Abstract For many years, a major question in sieve theory has been determining whether or not a sieve produces infinitely many values which are exactly two apart. In this paper, we will discuss a new result in sieve theory, which will give sufficient conditions for the existence of values which are exactly two apart. We will also show a direct application of this theorem on an existing sieve as well as detailing attempts to apply the theorem to the Sieve of Eratosthenes. iii Contents 1 Introduction 1 2 Preliminary Material 1 3 Sieves 5 3.1 The Sieve of Eratosthenes . 5 3.2 The Block Sieve . 9 3.3 The Finite Block Sieve . 12 3.4 The Sieve of Joseph Flavius . -
Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
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. -
Impartial Games
Combinatorial Games MSRI Publications Volume 29, 1995 Impartial Games RICHARD K. GUY In memory of Jack Kenyon, 1935-08-26 to 1994-09-19 Abstract. We give examples and some general results about impartial games, those in which both players are allowed the same moves at any given time. 1. Introduction We continue our introduction to combinatorial games with a survey of im- partial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81{116, and in ONAG [Conway 1976], particu- larly pp. 112{130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996], pp. ??{?? in this volume. An impartial game is one in which the set of Left options is the same as the set of Right options. We've noticed in the preceding article the impartial games = 0=0; 0 0 = 1= and 0; 0; = 2: {|} ∗ { | } ∗ ∗ { ∗| ∗} ∗ that were born on days 0, 1, and 2, respectively, so it should come as no surprise that on day n the game n = 0; 1; 2;:::; (n 1) 0; 1; 2;:::; (n 1) ∗ {∗ ∗ ∗ ∗ − |∗ ∗ ∗ ∗ − } is born. In fact any game of the type a; b; c;::: a; b; c;::: {∗ ∗ ∗ |∗ ∗ ∗ } has value m,wherem =mex a;b;c;::: , the least nonnegative integer not in ∗ { } the set a;b;c;::: . To see this, notice that any option, a say, for which a>m, { } ∗ This is a slightly revised reprint of the article of the same name that appeared in Combi- natorial Games, Proceedings of Symposia in Applied Mathematics, Vol. 43, 1991. Permission for use courtesy of the American Mathematical Society. -
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. -
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 -
What Can Be Said About the Number 13 Beyond the Fact That It Is a Prime Number? *
CROATICA CHEMICA ACTA CCACAA 77 (3) 447¿456 (2004) ISSN-0011-1643 CCA-2946 Essay What Can Be Said about the Number 13 beyond the Fact that It Is a Prime Number? * Lionello Pogliani,a Milan Randi},b and Nenad Trinajsti}c,** aDipartimento di Chimica, Universitá della Calabria, 879030 Rende (CS), Italy bDepartment of Mathematics and Computer Science, Drake University, Des Moines, Iowa 50311, USA cThe Rugjer Bo{kovi} Institute, P.O. Box 180, HR-10002 Zagreb, Croatia RECEIVED DECEMBER 2, 2003; REVISED MARCH 8, 2004; ACCEPTED MARCH 9, 2004 Key words The story of the number 13 that goes back to ancient Egypt is told. The mathematical signifi- art cance of 13 is briefly reviewed and 13 is discussed in various contexts, with special reference chemistry to the belief of many that this number is a rather unlucky number. Contrary examples are also history presented. Regarding everything, the number 13 appears to be a number that leaves no one in- literature different. number theory poetry science INTRODUCTION – religious [3 in the Christian faith, 5 in Islam]; – dramatic [3 and 9 by William Shakespeare (1564– »Numero pondere et mensura Deus omnia condidit.« 1616) in Macbeth]; Isaac Newton in 1722 [ 1 – literary 3 and 10 by Graham Greene (1904–1991) (Rozsondai and Rozsondai) in the titles of his successful short novels The The concept of number is one of the oldest and most Third Man and The Tenth Man; 5 and 9 by Dorothy useful concepts in the history of human race.2–5 People Leigh Sayers (1893–1957) in the titles of her nov- have been aware of the number concept from the very els The Five Red Herrings and The Nine Taylors. -
The Entropy Conundrum: a Solution Proposal
OPEN ACCESS www.sciforum.net/conference/ecea-1 Conference Proceedings Paper – Entropy The Entropy Conundrum: A Solution Proposal Rodolfo A. Fiorini 1,* 1 Politecnico di Milano, Department of Electronics, Information and Bioengineering, Milano, Italy; E- Mail: [email protected] * E-Mail: [email protected]; Tel.: +039-02-2399-3350; Fax: +039-02-2399-3360. Received: 11 September 2014 / Accepted: 11 October 2014 / Published: 3 November 2014 Abstract: In 2004, physicist Mark Newman, along with biologist Michael Lachmann and computer scientist Cristopher Moore, showed that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given 1-D signal is indistinguishable from blackbody radiation. Since many natural processes maximize the Gibbs- Boltzmann entropy, they should give rise to spectra indistinguishable from optimally efficient transmission. In 2008, computer scientist C.S. Calude and physicist K. Svozil proved that "Quantum Randomness" is not Turing computable. In 2013, academic scientist R.A. Fiorini confirmed Newman, Lachmann and Moore's result, creating analogous example for 2-D signal (image), as an application of CICT in pattern recognition and image analysis. Paradoxically if you don’t know the code used for the message you can’t tell the difference between an information-rich message and a random jumble of letters. This is an entropy conundrum to solve. Even the most sophisticated instrumentation system is completely unable to reliably discriminate so called "random noise" from any combinatorially optimized encoded message, which CICT called "deterministic noise". Entropy fundamental concept crosses so many scientific and research areas, but, unfortunately, even across so many different disciplines, scientists have not yet worked out a definitive solution to the fundamental problem of the logical relationship between human experience and knowledge extraction. -
Newsletter 91
Newsletter 9 1: December 2010 Introduction This is the final nzmaths newsletter for 2010. It is also the 91 st we have produced for the website. You can have a look at some of the old newsletters on this page: http://nzmaths.co.nz/newsletter As you are no doubt aware, 91 is a very interesting and important number. A quick search on Wikipedia (http://en.wikipedia.org/wiki/91_%28number%29) will very quickly tell you that 91 is: • The atomic number of protactinium, an actinide. • The code for international direct dial phone calls to India • In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar) • The ISBN Group Identifier for books published in Sweden. In more mathematically related trivia, 91 is: • the twenty-seventh distinct semiprime. • a triangular number and a hexagonal number, one of the few such numbers to also be a centered hexagonal number, and it is also a centered nonagonal number and a centered cube number. It is a square pyramidal number, being the sum of the squares of the first six integers. • the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): 91 = 6 3+(-5) 3 = 43+33. • the smallest positive integer expressible as a sum of six distinct squares: 91 = 1 2+2 2+3 2+4 2+5 2+6 2.