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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. -
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. -
Guidance Via Number 40: a Unique Stance M
Sci.Int.(Lahore),31(4),603-605, 2019 ISSN 1013-5316;CODEN: SINTE 8 603 GUIDANCE VIA NUMBER 40: A UNIQUE STANCE M. AZRAM 7-William St. Wattle Grove, WA 6107, USA. E-mail: [email protected] ABSTRACT: The number Forty (40) is a mysterious number that has a great significance in Mathematics, Science, astronomy, sports, spiritual traditions and many cultures. Historically, many things took place labelled with number 40. It has also been repeated many times in Islam. There must be some significantly importance associated with this magic number 40. Only Allah knows the complete answer. Since Muslims are enjoined to establish good governance through a just sociomoral order (or a state) wherein they could organise their individual and collective life in accordance with the teachings of the Qur’an and the Sunnah of the Prophet (SAW). I have gathered together some information to find some guideline via significant number 40 to reform our community by eradicating corruption, exploitation, injustice and evil etc. INTRODUCTION Muslims are enjoined to establish good governance through a just sociomoral order (or a state) wherein they could organise their individual and collective life in accordance with the teachings of the Qur’an and the Sunnah of the pentagonal pyramidal number[3] and semiperfect[4] .It is the expectation of the Qur’an that its number which have fascinated mathematics .(ﷺ) Prophet 3. 40 is a repdigit in base 3 (1111, i.e. 30 + 31 + 32 + 33) adherents would either reform the earth (by eradicating , Harshad number in base10 [5] and Stormer number[6]. corruption, exploitation, injustice and evil) or lay their lives 4. -
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. -
Large and Small Gaps Between Consecutive Niven Numbers
1 2 Journal of Integer Sequences, Vol. 6 (2003), 3 Article 03.2.5 47 6 23 11 Large and small gaps between consecutive Niven numbers Jean-Marie De Koninck1 and Nicolas Doyon D¶epartement de math¶ematiques et de statistique Universit¶e Laval Qu¶ebec G1K 7P4 Canada [email protected] [email protected] Abstract A positive integer is said to be a Niven number if it is divisible by the sum of its decimal digits. We investigate the occurrence of large and small gaps between consecutive Niven numbers. 1 Introduction A positive integer n is said to be a Niven number (or a Harshad number) if it is divisible by the sum of its (decimal) digits. For instance, 153 is a Niven number since 9 divides 153, while 154 is not. Niven numbers have been extensively studied; see for instance Cai [1], Cooper and Kennedy [2], Grundman [5] or Vardi [6]. Let N(x) denote the number of Niven numbers · x. Recently, De Koninck and Doyon proved [3], using elementary methods, that given any " > 0, x log log x x1¡" ¿ N(x) ¿ : log x Later, using complex variables as well as probabilistic number theory, De Koninck, Doyon and K¶atai [4] showed that x N(x) = (c + o(1)) ; (1) log x 1Research supported in part by a grant from NSERC. 1 where c is given by 14 c = log 10 ¼ 1:1939: (2) 27 In this paper, we investigate the occurrence of large gaps between consecutive Niven numbers. Secondly, denoting by T (x) the number of Niven numbers n · x such that n + 1 is also a Niven number, we prove that x log log x T (x) ¿ : (log x)2 We conclude by stating a conjecture. -
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”. -
Special Pythagorean Triangle in Relation with Pronic Numbers
International Journal of Mathematics Trends and Technology (IJMTT) – Volume 65 Issue 8 - August 2019 Special Pythagorean Triangle In Relation With Pronic Numbers S. Vidhyalakshmi1, T. Mahalakshmi2, M.A. Gopalan3 1,2,3(Department of Mathematics, Shrimati Indra Gandhi College, India) Abstract: 2* Area This paper illustrates Pythagorean triangles, where, in each Pythagorean triangle, the ratio Perimeter is a Pronic number. Keywords: Pythagorean triangles,Primitive Pythagorean triangle, Non primitive Pythagorean triangle, Pronic numbers. Introduction: It is well known that there is a one-to-one correspondence between the polygonal numbers and the sides of polygon. In addition to polygon numbers, there are other patterns of numbers namely Nasty numbers, Harshad numbers, Dhuruva numbers, Sphenic numbers, Jarasandha numbers, Armstrong numbers and so on. In particular, refer [1-17] for Pythagorean triangles in connection with each of the above special number patterns. The above results motivated us for searching Pythagorean triangles in connection with a new number pattern. This paper 2* Area illustrates Pythagorean triangles, where, in each Pythagorean triangle, the ratio is a Pronic number. Perimeter Method of Analysis: Let Tx, y, z be a Pythagorean triangle, where x 2pq , y p2 q2 , z p2 q2 , p q 0 (1) Denote the area and perimeter of Tx, y, z by A and P respectively. The mathematical statement of the problem is 2A nn 1 , pronic number of rank n (2) P qp q nn 1 (3) It is observed that (3) is satisfied by the following two sets of values of p and q: Set 1: p 2n 1, q n Set 2: p 2n 1, q n 1 However, there are other choices for p and q that satisfy (3). -
Eureka Issue 61
Eureka 61 A Journal of The Archimedeans Cambridge University Mathematical Society Editors: Philipp Legner and Anja Komatar © The Archimedeans (see page 94 for details) Do not copy or reprint any parts without permission. October 2011 Editorial Eureka Reinvented… efore reading any part of this issue of Eureka, you will have noticed The Team two big changes we have made: Eureka is now published in full col- our, and printed on a larger paper size than usual. We felt that, with Philipp Legner Design and Bthe internet being an increasingly large resource for mathematical articles of Illustrations all kinds, it was necessary to offer something new and exciting to keep Eu- reka as successful as it has been in the past. We moved away from the classic Anja Komatar Submissions LATEX-look, which is so common in the scientific community, to a modern, more engaging, and more entertaining design, while being conscious not to Sean Moss lose any of the mathematical clarity and rigour. Corporate Ben Millwood To make full use of the new design possibilities, many of this issue’s articles Publicity are based around mathematical images: from fractal modelling in financial Lu Zou markets (page 14) to computer rendered pictures (page 38) and mathemati- Subscriptions cal origami (page 20). The Showroom (page 46) uncovers the fundamental role pictures have in mathematics, including patterns, graphs, functions and fractals. This issue includes a wide variety of mathematical articles, problems and puzzles, diagrams, movie and book reviews. Some are more entertaining, such as Bayesian Bets (page 10), some are more technical, such as Impossible Integrals (page 80), or more philosophical, such as How to teach Physics to Mathematicians (page 42). -
ON FINDING the SMALLEST HAPPY NUMBERS of ANY HEIGHTS 2 Where B(0) = 0 in Base B
ON FINDING THE SMALLEST HAPPY NUMBERS OF ANY HEIGHTS GABRIEL LAPOINTE Abstract. This paper focuses on finding the smallest happy number for each height in any numerical base. Using the properties of the height, we deduce a recursive relationship between the smallest happy number and the height where the initial height is function of the numerical base. With the usage of the recursive relationship, we build an algorithm that exploits the properties of the height in order to find all of those smallest happy numbers with unknown height. However, with the modular arithmetic, we conclude on an equation that calculates the smallest happy numbers based on known heights for binary and ternary bases. 1. Introduction Researches have been done on smallest happy numbers mainly in decimal base for heights 0 to 12 using the modular arithmetic as shown in theorems 2 to 4 [2] and theorem 3 [6]. Moreover, there is an algorithm that searches for the smallest happy number of an unknown height as shown in [6]. In this paper, we give and describe this algorithm usable for any numerical base B 2. Instead of continuing to search for the smallest happy numbers for any height≥ greater than 12 in decimal base, we are interested on the binary and ternary bases. Small numerical bases may give hints on an equation or an efficient algorithm to obtain the smallest happy number of any height for larger numerical bases. Let x > 0 be an integer. In the numerical base B 2, x can be written as a unique sum ≥ L(x)−1 i x = xiB , (1) i=0 X where the positive integers x0,...,xL(x)−1 are the digits of x and xj = 0 for all j arXiv:1904.12032v2 [math.NT] 7 May 2019 ≥ L(x). -
Fermat Pseudoprimes
1 TWO HUNDRED CONJECTURES AND ONE HUNDRED AND FIFTY OPEN PROBLEMS ON FERMAT PSEUDOPRIMES (COLLECTED PAPERS) Education Publishing 2013 Copyright 2013 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: 9781599732572 ISBN: 978-1-59973-257-2 1 INTRODUCTION Prime numbers have always fascinated mankind. For mathematicians, they are a kind of “black sheep” of the family of integers by their constant refusal to let themselves to be disciplined, ordered and understood. However, we have at hand a powerful tool, insufficiently investigated yet, which can help us in understanding them: Fermat pseudoprimes. It was a night of Easter, many years ago, when I rediscovered Fermat’s "little" theorem. Excited, I found the first few Fermat absolute pseudoprimes (561, 1105, 1729, 2465, 2821, 6601, 8911…) before I found out that these numbers are already known. Since then, the passion for study these numbers constantly accompanied me. Exceptions to the above mentioned theorem, Fermat pseudoprimes seem to be more malleable than prime numbers, more willing to let themselves to be ordered than them, and their depth study will shed light on many properties of the primes, because it seems natural to look for the rule studying it’s exceptions, as a virologist search for a cure for a virus studying the organisms that have immunity to the virus. -
Techsparx Java Tuitions
TechSparxJavaTuitionsTechSparxJavaJava Question Bank TuitionsTechSparxJavaTuitionsTechSp arxJavaTuitionsTechSparxJavaTuitions TechSparxJavaTuitionsTechSparxJavaTechSparx Java Tuitions “Better than a thousand days of diligent study is one day with a Great Teacher” TuitionsTechSparxJavaTuitionsTechSp Java Question Bank (version 5.3) arxJavaTuitionsTechSparxJavaTuitionshttp://techsparx.webs.com/ Saravanan.G TechSparxJavaTuitionsTechSparxJava TuitionsTech SparxJavaTuitionsTechSp arxJavaTuitionsTechSparxJavaTuitions TechSparxJavaTuitionsTechSparxJava TuitionsTechSparxJavaTuitionsTechSp arxJavaTuitionsTechSparxJavaTuitions TechSparxJavaTuitionsTechSparxJava TuitionsTechSparxJavaTuitionsTechSp arxJavaTuitionsTechSparxJavaTuitions TechSparxJavaTuitionsTechSparxJava TuitionsTechSparxJavaTuitionsTechSp Contents Modularization ........................................................................................................................................ 3 Most Simplest ......................................................................................................................................... 4 Decision Making Statements ................................................................................................................... 5 Switch Statement .................................................................................................................................... 6 Looping Constructs .................................................................................................................................