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Further Generalizations of Happy Numbers
Further Generalizations of Happy Numbers E. S. Williams June 15, 2016 Abstract In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple in [2] and establish further results, not given in that work. Then we construct a similar function expanding the definition of happy numbers to negative integers. Using this function, we compute and prove results extending those regarding higher powers and sequences of consecutive happy numbers that El-Sidy and Siksek and Grundman and Teeple proved in [1], [2] and [3] to negative integers. Finally, we consider a variety of special cases, in which the existence of certain fixed points and cycles of infinite families of generalized happy functions can be proven. 1 Introduction Happy numbers have been studied for many years, although the origin of the concept is unclear. Consider the sum of the square of the digits of an arbitrary positive integer. If repeating the process of taking the sums of the squares of the digits of an integer eventually gets us 1, then that integer is happy. This paper first provides a brief overview of traditional happy numbers, then describes existing work on generalizations of the concept to other bases and higher powers. We extend some earlier results for the cubic case to the quartic case. We then extend the happy function S to negative integers, constructing a function Q : Z ! Z which agrees with S over Z+ for this purpose. We then generalize Q to various bases and higher powers. As is traditional in 1 the study of special numbers, we consider consecutive sequences of happy numbers, and generalize this study to Q. -
M-Gonal Number± N = Nasty Number, N =
International Journal of Research in Advent Technology, Vol.6, No.11, November 2018 E-ISSN: 2321-9637 Available online at www.ijrat.org M-Gonal number n = Nasty number, n = 1,2 Dr.K.Geetha Assistant Professor, Department of Mathematics, Dr.Umayal Ramanathan College for Women, Karaikudi. [email protected] Abstract: This paper concerns with study of obtaining the ranks of M-gonal numbers along with their recurrence relation such that M-gonal number n = Nasty number where n = 1,2. Key Words: Triangular number, Pentagonal number, Hexagonal number, Octagonal number and Dodecagonal number 1. INTRODUCTION may refer [3-7]. In this communication an attempt is Numbers exhibit fascinating properties, they form made to obtain the ranks of M-gonal numbers like patterns and so on [1]. In [3] the ranks of Triangular, Triangular, Pentagonal, Hexagonal, Octagonal and Pentagonal, Heptagonal, Nanogonal and Tridecagonal Dodecagonal numbers such that each of these M-gonal numbers such that each of these M-gonal number - number n = Nasty number, n = 1,2. Also the 2 = a perfect square and the ranks of Pentagonal and recurrence relations satisfied by the ranks of each of Heptagonal such that each of these M-gonal number + these M-gonal numbers are presented. 2= a perfect square are obtained. In this context one 2. METHOD OF ANALYSIS 2.1.Pattern 1: Let the rank of the nth Triangular number be A, then the identity 2 Triangular number – 2 = 6 x (1) is written as y22 48x 9 (2) where y 2A 1 (3) whose initial solution is x30 , y0 21 (4) Let xyss, be the general solution of the Pellian 22 yx48 1 (5) 1 ss11 where xs 7 48 7 48 2 48 1 ss11 ys 7 48 7 48 , s 0,1,... -
[Math.NT] 10 May 2005
Journal de Th´eorie des Nombres de Bordeaux 00 (XXXX), 000–000 Restriction theory of the Selberg sieve, with applications par Ben GREEN et Terence TAO Resum´ e.´ Le crible de Selberg fournit des majorants pour cer- taines suites arithm´etiques, comme les nombres premiers et les nombres premiers jumeaux. Nous demontrons un th´eor`eme de restriction L2-Lp pour les majorants de ce type. Comme ap- plication imm´ediate, nous consid´erons l’estimation des sommes exponentielles sur les k-uplets premiers. Soient les entiers posi- tifs a1,...,ak et b1,...,bk. Ecrit h(θ) := n∈X e(nθ), ou X est l’ensemble de tous n 6 N tel que tousP les nombres a1n + b1,...,akn+bk sont premiers. Nous obtenons les bornes sup´erieures pour h p T , p > 2, qui sont (en supposant la v´erit´ede la con- k kL ( ) jecture de Hardy et Littlewood sur les k-uplets premiers) d’ordre de magnitude correct. Une autre application est la suivante. En utilisant les th´eor`emes de Chen et de Roth et un «principe de transf´erence », nous demontrons qu’il existe une infinit´ede suites arithm´etiques p1 < p2 < p3 de nombres premiers, telles que cha- cun pi + 2 est premier ou un produit de deux nombres premier. Abstract. The Selberg sieve provides majorants for certain arith- metic sequences, such as the primes and the twin primes. We prove an L2–Lp restriction theorem for majorants of this type. An im- mediate application is to the estimation of exponential sums over prime k-tuples. -
A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics
ON THE FORM OF MERSENNE NUMBERS PREDRAG TERZICH Abstract. We present a theorem concerning the form of Mersenne p numbers , Mp = 2 − 1 , where p > 3 . We also discuss a closed form expression which links prime numbers and natural logarithms . 1. Introduction. The friar Marin Mersenne (1588 − 1648), a member of the order of Minims, was a philosopher, mathematician, and scientist who wrote on music, mechanics, and optics . A Mersenne number is a number of p the form Mp = 2 − 1 where p is an integer , however many authors prefer to define a Mersenne number as a number of the above form but with p restricted to prime values . In this paper we shall use the second definition . Because the tests for Mersenne primes are relatively simple, the record largest known prime number has almost always been a Mersenne prime . 2. A theorem and closed form expression Theorem 2.1. Every Mersenne number Mp , (p > 3) is a number of the form 6 · k · p + 1 for some odd possitive integer k . Proof. In order to prove this we shall use following theorems : Theorem 2.2. The simultaneous congruences n ≡ n1 (mod m1) , n ≡ n2 (mod m2), .... ,n ≡ nk (mod mk) are only solvable when ni = nj (mod gcd(mi; mj)) , for all i and j . i 6= j . The solution is unique modulo lcm(m1; m2; ··· ; mk) . Proof. For proof see [1] Theorem 2.3. If p is a prime number, then for any integer a : ap ≡ a (mod p) . Date: January 4 , 2012. 1 2 PREDRAG TERZICH Proof. -
The Pseudoprimes to 25 • 109
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a. -
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. -
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 . -
Simple Rules for Incorporating Design Art Into Penrose and Fractal Tiles
Bridges 2012: Mathematics, Music, Art, Architecture, Culture Simple Rules for Incorporating Design Art into Penrose and Fractal Tiles San Le SLFFEA.com [email protected] Abstract Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable M.C. Escher-like image that works esthetically as well as functionally requires resolving incongruencies at a tile’s edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical objects to which he never applied his artistry including Penrose Tilings and fractals. In this paper, we show that the rules of creating a traditional tile extend to these objects as well. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others. 1 1 Introduction M.C. Escher was the most prominent artist working with tessellations and space filling. Forty years after his death, his creations are still foremost in people’s minds in the field of tiling art. One of the reasons Escher continues to hold such a monopoly in this specialty are the unique challenges that come with creating Escher type designs inside a tessellation[15]. When an image is drawn into a tile and extends to the tile’s edge, it introduces incongruencies which are resolved by continuously aligning and refining the image. This is particularly true when the image consists of the lizards, fish, angels, etc. which populated Escher’s tilings because they do not have the 4-fold rotational symmetry that would make it possible to arbitrarily rotate the image ± 90, 180 degrees and have all the pieces fit[9]. -
1. Introduction
Journal of Combinatorics and Number Theory ISSN 1942-5600 Volume 2, Issue 3, pp. 65-77 c 2010 Nova Science Publishers, Inc. CYCLES AND FIXED POINTS OF HAPPY FUNCTIONS Kathryn Hargreaves∗ and Samir Sikseky Mathematics Institute, University of Warwick Coventry, CV4 7AL, United Kingdom Received October 8, 2009; Accepted September 22, 2010 Abstract Let N = f1; 2; 3; · · · g denote the natural numbers. Given integers e ≥ 1 and Pn i b ≥ 2, let x = i=0 aib with 0 ≤ ai ≤ b − 1 (thus ai are the digits of x in base b). We define the happy function Se;b : N −! N by n X e Se;b(x) = ai : i=0 r A positive integer x is then said to be (e; b)-happy if Se;b(x) = 1 for some r ≥ 0, otherwise we say it is (e; b)-unhappy. In this paper we investigate the cycles and fixed points of the happy functions Se;b. We give an upper bound for the size of elements belonging to the cycles of Se;b. We also prove that the number of fixed points of S2;b is ( b2+1 2 τ 2 − 1 if b is odd, τ(b2 + 1) − 1 if b is even, where τ(m) denotes the number of positive divisors of m. We use our results to determine the densities of happy numbers in a small handful of cases. 2000 Mathematics Subject Classification: Primary 11A63; Secondary 11B05. 1. Introduction Guy [G] introduces Problem E34 Happy numbers as follows: Reg. Allenby’s daughter came home from school in Britain with the concept of happy numbers. -
Monthly Science and Maths Magazine 01
1 GYAN BHARATI SCHOOL QUEST….. Monthly Science and Mathematics magazine Edition: DECEMBER,2019 COMPILED BY DR. KIRAN VARSHA AND MR. SUDHIR SAXENA 2 IDENTIFY THE SCIENTIST She was an English chemist and X-ray crystallographer who made contributions to the understanding of the molecular structures of DNA , RNA, viruses, coal, and graphite. She was never nominated for a Nobel Prize. Her work was a crucial part in the discovery of DNA, for which Francis Crick, James Watson, and Maurice Wilkins were awarded a Nobel Prize in 1962. She died in 1958, and during her lifetime the DNA structure was not considered as fully proven. It took Wilkins and his colleagues about seven years to collect enough data to prove and refine the proposed DNA structure. RIDDLE TIME You measure my life in hours and I serve you by expiring. I’m quick when I’m thin and slow when I’m fat. The wind is my enemy. Hard riddles want to trip you up, and this one works by hitting you with details from every angle. The big hint comes at the end with the wind. What does wind threaten most? I have cities, but no houses. I have mountains, but no trees. I have water, but no fish. What am I? This riddle aims to confuse you and get you to focus on the things that are missing: the houses, trees, and fish. 3 WHY ARE AEROPLANES USUALLY WHITE? The Aeroplanes might be having different logos and decorations. But the colour of the aeroplane is usually white.Painting the aeroplane white is most practical and economical. -
WGLT Program Guide, March-April, 2012
Illinois State University ISU ReD: Research and eData WGLT Program Guides Arts and Sciences Spring 3-1-2012 WGLT Program Guide, March-April, 2012 Illinois State University Follow this and additional works at: https://ir.library.illinoisstate.edu/wgltpg Recommended Citation Illinois State University, "WGLT Program Guide, March-April, 2012" (2012). WGLT Program Guides. 241. https://ir.library.illinoisstate.edu/wgltpg/241 This Book is brought to you for free and open access by the Arts and Sciences at ISU ReD: Research and eData. It has been accepted for inclusion in WGLT Program Guides by an authorized administrator of ISU ReD: Research and eData. For more information, please contact [email protected]. You'll be able to count on GLT's own award-winning news team because we will Seven is a not only a prime number but also a double Mersenne prime, have the resources they need to continue a Newman-Shanks-Williams prime, a Woodall prime, a factorial prime, providing you with top quality local a lucky prime, a safe prime (the only Mersenne safe prime) and a "happy and regional news. prime," or happy number. Whether you love our top shelf jazz According to Sir Isaac Newton, a rainbow has seven colors. service or great weekend blues - or both - There are seven deadly sins, seven dwarves, seven continents, seven you'll get what you want when you tune in ancient wonders of the world, seven chakras and seven days in the week. because we'll have the tools and the talent to bring you the music you love. -
Single Digits
...................................single digits ...................................single digits In Praise of Small Numbers MARC CHAMBERLAND Princeton University Press Princeton & Oxford Copyright c 2015 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved The second epigraph by Paul McCartney on page 111 is taken from The Beatles and is reproduced with permission of Curtis Brown Group Ltd., London on behalf of The Beneficiaries of the Estate of Hunter Davies. Copyright c Hunter Davies 2009. The epigraph on page 170 is taken from Harry Potter and the Half Blood Prince:Copyrightc J.K. Rowling 2005 The epigraphs on page 205 are reprinted wiht the permission of the Free Press, a Division of Simon & Schuster, Inc., from Born on a Blue Day: Inside the Extraordinary Mind of an Austistic Savant by Daniel Tammet. Copyright c 2006 by Daniel Tammet. Originally published in Great Britain in 2006 by Hodder & Stoughton. All rights reserved. Library of Congress Cataloging-in-Publication Data Chamberland, Marc, 1964– Single digits : in praise of small numbers / Marc Chamberland. pages cm Includes bibliographical references and index. ISBN 978-0-691-16114-3 (hardcover : alk. paper) 1. Mathematical analysis. 2. Sequences (Mathematics) 3. Combinatorial analysis. 4. Mathematics–Miscellanea. I. Title. QA300.C4412 2015 510—dc23 2014047680 British Library