DOCSLIB.ORG

A Counting Function for the Sequence of Perfect Powers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Counting Function for the Sequence of Perfect Powers A Counting Function for the Sequence of Perfect Powers M A Nyblom Scho ol of Mathematics and Geospatial Science RMIT University GPO Box V Melb ourne Victorial Australia email michaelnyblomrmiteduau Intro duction n A natural numb er of the form m where m is a p ositiveinteger and n is called a p erfect power Unsolved problems concerning the set of p erfect p owers ab ound throughout muchof numb er theory The most famous of these is known as the Catalan conjecture which states that the only p erfect p owers which dier by unity are the integers and It is of interest to note that this particular problem has only recently b een solved using rather deep results from the theory of cyclotomic elds see The set of p erfect p owers can naturally b e arranged into an increasing sequence of distinct integers in which those p erfect powers expressible with dieren texponents are treated as a single element of the sequence The rst few terms of this sequence of p erfect p owers without duplication are and is listed in the OnLine Encyclop edia of Integer Sequences under Sloane A The sequence in has many prop erties one b eing that the innite sum of its recipro cals is convergent see a clear indication of the scarcity of the p erfect powers amongst the set of natural numb ers This latter fact is naturally reected in the well known result that the sequence of p erfect p owers has zero asymptotic densitythat is if N x denotes the number of elements of less than a p ositivereal x then lim N xx In view of this result x one may question what is the precise nature of the growth rate of the counting function N x in particular can an asymptotic estimate for N x b e found We shall establish sucha p x as x distributional result for the sequence of p erfect p owers by proving that N x As will b e seen this asymptotic formula can b e interpreted as stating that the p erfect squares dominate the count of the sequence elements in as x To contrast the main result we shall in addition develop a closedform expression for N x using elementary sieve metho ds As will be seen this formula is some what reminiscent to Legendres counting function for p x x In what follows we denote the integer part of the number of primes in the interval x by bxc An Asymptotic Formula To help establish the main results of this pap er we shall rst need to formally intro duce the following family of sets Denition Suppose x and n N nf g thenletA x denote the set of perfect powers n n n having exponent n and which are less than or equal to x that is A xfk k N k xg n Wenow establish the asymptotic formula for the counting function N x Theorem If N x denotes the number of sequence elements of that are less than or p equal to x then N x x as x Pro of The rst step of the argument will b e to obtain upp er and lower functional b ounds for N x Assuming without loss of generality that x observe A x for each n n N nf g but for n suciently large A xnfg Dening the auxiliary function n M xmaxfn N nf g A xnfg g we clearly see M x as A xnfg and n S M x that N x is equal to the number of elements of the set A A x Furthermore from n n blog xc blog xc the inequality x it is immediately deduced that M x blog xc blog xc Since for large x the family of sets fA xg are not mutually disjoint it follows that n n blog xc X N xjAj jA xj n n and since A x A one also has jA xjjAj N x n n Now as A x there must exist a largest integer m such that m xm n p n xm that By taking the nth ro ot through the previous inequalitywe deduce m p p n n is m b xc and so A xmust contain b xc elements Consequently and together n yields that blog xc X p p n xcN x b xc b n Using the upp er and lower b ounds in we can establish required the asymptotic estimate p x observe for large x the following train of inequalities for N x as follows Dividing by p p p p p blog xc blog xc n n X X b xc N x b xc xc b xc x b p p p p p p x x x x x x n n p p blog xc X xc x b p p x x n p b blog xc xc p p x x Via an application of LHopitals rule it is easily seen that log x blog xc p p x x as x moreover by recalling lim bxcx we nally deduce from that x p N x x asx p Remark Since the number of perfect squares less than or equal to x is given by b xc p p xc x we can interpret Theorem as stating that the perfect squares dominate and as b the count of the sequence elements of as x An Exact Formula One of the earliest known sieve metho ds was a simple eective pro cedure for nding all prime numb ers up to a certain bound x This pro cedure whichinvolves the systematic deletion of p all multiples of primes less than or equal to x was captured succinctly by Legendre using a theoretical analog of the sifting pro cess known to day as the InclusionExclusion Principal to study the prime counting function x jfp x p a prime gj His metho d led to an p exact formula for the number of primes in the interval x x in particular if denotes the Mobius function then k j X p x x d x d djP x Q p where the sum is taken over all divisors of P p see pg In this section we x p x shall employ the same elementary sieve metho d of Legendre to establish an exact formula for the counting function N x which is similar in form to We b egin with a technical lemma for the sets of Denition Lemma For any set of m positive integers fn n g al l greater than unity m m A xA x n n n i m i where n n denotes the least common multiple of the m integers n n m m g Pro of We begin by demonstrating that A x A xA x for any n m N nf n m nm which is the base step of our inductive argument Nowsincenjn mandmjn manynumber nm of the form k where k N can b e rewritten as a p erfect p ower having an exp onent n and n m m thus A x A x A x Let s A x A x with s then s k k n m n m nm nm for some k k N nf g W e have to pro duce a k N nf g such that s k As n m r k k both k and k must have the same prime divisors Writing k p p p r r n m and k p p p we deduce from the equality k k that n m for each r i i i r Consequently njn and mjn and so n n m for some N Thus i i i i i r nm s k where k p p p which establishes that A x A x A x r n m nm Now supp ose for m the set identity in holds for an arbitrary set of m p ositiveintegers n n n n n n observe fn n g all greater than unity Then as m m m m from the inductive assumption and the base step that m m A x A x A x A x A x n n n n n n m m i i m i i A x n n n m m A x n n m Hence holds for m arbitrary p ositiv eintegers greater than unity and so the result is established by the principal of mathematical induction Theorem If x then the counting function for the sequence in is given by the explicit expression X d N xbxc dbx c djP x Q where the sum is taken over al l divisors of P p x pblog xc Pro of We b egin by establishing a slight reformulation for the set A of Theorem Recall blog xc ing that A A x we claim if p p are the rst m primes less than or equal n m n to blog xc then in fact A B where m x B A p r r x is included in the The inclusion B A follows automatically by denition as eachsetA p k union of sets which form A To establish the reverse inclusion A B rst observe that as p p represent the complete list of primes less than or equal to blog xc every integer m n f blog xcg must b e divisible by at least one of these primes since otherwise by log xc and the fundamental theorem of arithmetic n would be divisible by a prime p b n so n blog xc a contradiction Consequentlyifgiven any s A x then s k and one n p r x and so A may write n p for some r f mg and N Thus s k p r r every elementofA is contained in the set B m Now N xjAj jB j and since for x large the family of sets fA x g are not mutually p i i disjointwe deduce from an application of the InclusionExclusion Principal applied to the set B that m X X k N x jA x A xj p p i i k i i m k k where the expression i i m indicates that the sum is taken over all ordered k k element subsets fi i g of the set f mg As the least common multiple of the k k prime num bers p p is clearly the pro duct d p p p observe from Lemma i i i i i k k d c noting here wehave again used the fact that jA x A xj jA xj bx p p d i i k Q p n that the number of elements in the set A x is b xc Dening P p we see n x pblog xc m that for each k f mg the inner summation in consists of adding terms of k d c where d p p p is a divisor of P having k distinct prime factors the form bx i i i x k k Consequently as p p p the double summation in must sum terms of the i i i k d form dbx c over all divisors dwhere d of P excluding d Finally by recalling that x we deduce that the right hand side of reduces to the right hand side of Remark An immediate consequence of Theorem is that the number of nonperfect P d c dbx powers less than or equal to x is equal to djP x Numerical Example We examine nowhow the explicit expression for N x in can b e practically implemented to compute the numb er of p erfect p owers less than or equal to a given large p ositive real x For notational convenience let the inner summation of b e denoted by k j X p p i i k S x x k i i m k p p i i k Observe that in order to evaluate each S x one must sum the terms bx c over k those subscripts i i whose values are chosen from the ordered k element subsets k m of f mg consequently the numb er of summands is Thus on rst acquaintance
Load more

Recommended publications
  • 81684-1B SP.Pdf
    INTERMEDIATE ALGEBRA The Why and the How First Edition Mathew Baxter Florida Gulf Coast University Bassim Hamadeh, CEO and Publisher Jess Estrella, Senior Graphic Designer Jennifer McCarthy, Acquisitions Editor Gem Rabanera, Project Editor Abbey Hastings, Associate Production Editor Chris Snipes, Interior Designer Natalie Piccotti, Senior Marketing Manager Kassie Graves, Director of Acquisitions Jamie Giganti, Senior Managing Editor Copyright © 2018 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of Cognella, Inc. For inquiries regarding permissions, translations, foreign rights, audio rights, and any other forms of reproduction, please contact the Cognella Licensing Department at [email protected]. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Cover image copyright © Depositphotos/agsandrew. Printed in the United States of America ISBN: 978-1-5165-0303-2 (pbk) / 978-1-5165-0304-9 (br) iii |  CONTENTS 1 The Real Number System 1 RealNumbersandTheirProperties 2:1.1 RealNumbers 2 PropertiesofRealNumbers 6 Section1.1Exercises 9 TheFourBasicOperations 11:1.2 AdditionandSubtraction 11 MultiplicationandDivision
    [Show full text]
  • On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
    mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer).
    [Show full text]
  • The Unique Natural Number Set and Distributed Prime Numbers
    mp Co uta & tio d n e a li Jabur and Kushnaw, J Appl Computat Math 2017, 6:4 l p M p a Journal of A t h DOI: 10.4172/2168-9679.1000368 f e o m l a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access The Unique Natural Number Set and Distributed Prime Numbers Alabed TH1* and Bashir MB2 1Computer Science, Shendi University, Shendi, Sudan 2Shendi University, Shendi, Sudan Abstract The Natural number set has a new number set behind three main subset: odd, even and prime which are unique number set. It comes from a mathematical relationship between the two main types even and odd number set. It consists of prime and T-semi-prime numbers set. However, finding ways to understand how prime number was distributed was ambiguity and it is impossible, this new natural set show how the prime number was distributed and highlight focusing on T-semi-prime number as a new sub natural set. Keywords: Prime number; T-semi-prime number; Unique number purely mathematical sciences and it has been used prime number set within theoretical physics to try and produce a theory that ties together prime numbers with quantum chaos theory [3]. However, researchers Introduction were made a big effort to find algorithm which can describe how prime The Unique natural number set is sub natural number set which number are distributed, still distribution of prime numbers throughout consist of prime and T-semi-prime numbers. T-semi-prime is natural a list of integers leads to the emergence of many unanswered and number divided by one or more prime numbers.
    [Show full text]
  • New Formulas for Semi-Primes. Testing, Counting and Identification
    New Formulas for Semi-Primes. Testing, Counting and Identification of the nth and next Semi-Primes Issam Kaddouraa, Samih Abdul-Nabib, Khadija Al-Akhrassa aDepartment of Mathematics, school of arts and sciences bDepartment of computers and communications engineering, Lebanese International University, Beirut, Lebanon Abstract In this paper we give a new semiprimality test and we construct a new formula for π(2)(N), the function that counts the number of semiprimes not exceeding a given number N. We also present new formulas to identify the nth semiprime and the next semiprime to a given number. The new formulas are based on the knowledge of the primes less than or equal to the cube roots 3 of N : P , P ....P 3 √N. 1 2 π( √N) ≤ Keywords: prime, semiprime, nth semiprime, next semiprime 1. Introduction Securing data remains a concern for every individual and every organiza- tion on the globe. In telecommunication, cryptography is one of the studies that permits the secure transfer of information [1] over the Internet. Prime numbers have special properties that make them of fundamental importance in cryptography. The core of the Internet security is based on protocols, such arXiv:1608.05405v1 [math.NT] 17 Aug 2016 as SSL and TSL [2] released in 1994 and persist as the basis for securing dif- ferent aspects of today’s Internet [3]. The Rivest-Shamir-Adleman encryption method [4], released in 1978, uses asymmetric keys for exchanging data. A secret key Sk and a public key Pk are generated by the recipient with the following property: A message enciphered Email addresses: [email protected] (Issam Kaddoura), [email protected] (Samih Abdul-Nabi) 1 by Pk can only be deciphered by Sk and vice versa.
    [Show full text]
  • Arxiv:Math/0201082V1
    2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e.
    [Show full text]
  • 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.
    [Show full text]
  • Appendix a Tables of Fermat Numbers and Their Prime Factors
    Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard
    [Show full text]
  • William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri [email protected]
    #A3 INTEGERS 16 (2016) EVERY NATURAL NUMBER IS THE SUM OF FORTY-NINE PALINDROMES William D. Banks Department of Mathematics, University of Missouri, Columbia, Missouri [email protected] Received: 9/4/15, Accepted: 1/4/16, Published: 1/21/16 Abstract It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes. 1. Statement of Result Let N .= 0, 1, 2, . denote the set of natural numbers (including zero). Every { } number n N has a unique decimal representation of the form 2 L 1 − j n = 10 δj, (1) j=0 X where each δj belongs to the digit set .= 0, 1, 2, . , 9 , D { } and the leading digit δL 1 is nonzero whenever L 2. In what follows, we use − ≥ diagrams to illustrate the ideas; for example, n = δL 1 δ1 δ0 − · · · represents the relation (1). The integer n is said to be a decimal palindrome if its decimal digits satisfy the symmetry condition δj = δL 1 j (0 j < L). − − Denoting by the collection of all decimal palindromes in N, the aim of this note P is to show that is an additive basis for N. P INTEGERS: 16 (2016) 2 Theorem 1. The set of decimal palindromes is an additive basis for the natural P numbers N. Every natural number is the sum of forty-nine (possibly zero) decimal palindromes. The proof is given in the next section. It is unlikely that the second statement is optimal; a refinement of our method may yield an improvement.
    [Show full text]
  • Fixed Points of Certain Arithmetic Functions
    FIXED POINTS OF CERTAIN ARITHMETIC FUNCTIONS WALTER E. BECK and RUDOLPH M. WAJAR University of Wisconsin, Whitewater, Wisconsin 53190 IWTRODUCTIOW Perfect, amicable and sociable numbers are fixed points of the arithemetic function L and its iterates, L (n) = a(n) - n, where o is the sum of divisor's function. Recently there have been investigations into functions differing from L by 1; i.e., functions L+, Z.,, defined by L + (n)= L(n)± 1. Jerrard and Temperley [1] studied the existence of fixed points of L+ and /._. Lai and Forbes [2] conducted a computer search for fixed points of (LJ . For earlier references to /._, see the bibliography in [2]. We consider the analogous situation using o*, the sum of unitary divisors function. Let Z.J, Lt, be arithmetic functions defined by L*+(n) = o*(n)-n±1. In § 1, we prove, using parity arguments, that L* has no fixed points. Fixed points of iterates of L* arise in sets where the number of elements in the set is equal to the power of/.* in question. In each such set there is at least one natural number n such that L*(n) > n. In § 2, we consider conditions n mustsatisfy to enjoy the inequality and how the inequality acts under multiplication. In particular if n is even, it is divisible by at least three primes; if odd, by five. If /? enjoys the inequality, any multiply by a relatively prime factor does so. There is a bound on the highest power of n that satisfies the inequality. Further if n does not enjoy the inequality, there are bounds on the prime powers multiplying n which will yield the inequality.
    [Show full text]
  • On Some New Arithmetic Functions Involving Prime Divisors and Perfect Powers
    On some new arithmetic functions involving prime divisors and perfect powers. Item Type Article Authors Bagdasar, Ovidiu; Tatt, Ralph-Joseph Citation Bagdasar, O., and Tatt, R. (2018) ‘On some new arithmetic functions involving prime divisors and perfect powers’, Electronic Notes in Discrete Mathematics, 70, pp.9-15. doi: 10.1016/ j.endm.2018.11.002 DOI 10.1016/j.endm.2018.11.002 Publisher Elsevier Journal Electronic Notes in Discrete Mathematics Download date 30/09/2021 07:19:18 Item License http://creativecommons.org/licenses/by/4.0/ Link to Item http://hdl.handle.net/10545/623232 On some new arithmetic functions involving prime divisors and perfect powers Ovidiu Bagdasar and Ralph Tatt 1,2 Department of Electronics, Computing and Mathematics University of Derby Kedleston Road, Derby, DE22 1GB, United Kingdom Abstract Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers ij obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences.
    [Show full text]
  • Problem Solving and Recreational Mathematics
    Problem Solving and Recreational Mathematics Paul Yiu Department of Mathematics Florida Atlantic University Summer 2012 Chapters 1–44 August 1 Monday 6/25 7/2 7/9 7/16 7/23 7/30 Wednesday 6/27 *** 7/11 7/18 7/25 8/1 Friday 6/29 7/6 7/13 7/20 7/27 8/3 ii Contents 1 Digit problems 101 1.1 When can you cancel illegitimately and yet get the cor- rectanswer? .......................101 1.2 Repdigits.........................103 1.3 Sortednumberswithsortedsquares . 105 1.4 Sumsofsquaresofdigits . 108 2 Transferrable numbers 111 2.1 Right-transferrablenumbers . 111 2.2 Left-transferrableintegers . 113 3 Arithmetic problems 117 3.1 AnumbergameofLewisCarroll . 117 3.2 Reconstruction of multiplicationsand divisions . 120 3.2.1 Amultiplicationproblem. 120 3.2.2 Adivisionproblem . 121 4 Fibonacci numbers 201 4.1 TheFibonaccisequence . 201 4.2 SomerelationsofFibonaccinumbers . 204 4.3 Fibonaccinumbersandbinomialcoefficients . 205 5 Counting with Fibonacci numbers 207 5.1 Squaresanddominos . 207 5.2 Fatsubsetsof [n] .....................208 5.3 Anarrangementofpennies . 209 6 Fibonacci numbers 3 211 6.1 FactorizationofFibonaccinumbers . 211 iv CONTENTS 6.2 TheLucasnumbers . 214 6.3 Countingcircularpermutations . 215 7 Subtraction games 301 7.1 TheBachetgame ....................301 7.2 TheSprague-Grundysequence . 302 7.3 Subtraction of powers of 2 ................303 7.4 Subtractionofsquarenumbers . 304 7.5 Moredifficultgames. 305 8 The games of Euclid and Wythoff 307 8.1 ThegameofEuclid . 307 8.2 Wythoff’sgame .....................309 8.3 Beatty’sTheorem . 311 9 Extrapolation problems 313 9.1 Whatis f(n + 1) if f(k)=2k for k =0, 1, 2 ...,n? . 313 1 9.2 Whatis f(n + 1) if f(k)= k+1 for k =0, 1, 2 ...,n? .
    [Show full text]
  • On Product Partitions of Integers
    Canad. Math. Bull.Vol. 34 (4), 1991 pp. 474-479 ON PRODUCT PARTITIONS OF INTEGERS V. C. HARRIS AND M. V. SUBBARAO ABSTRACT. Let p*(n) denote the number of product partitions, that is, the number of ways of expressing a natural number n > 1 as the product of positive integers > 2, the order of the factors in the product being irrelevant, with p*(\ ) = 1. For any integer d > 1 let dt = dxll if d is an /th power, and = 1, otherwise, and let d = 11°^ dj. Using a suitable generating function for p*(ri) we prove that n^,, dp*(n/d) = np*inK 1. Introduction. The well-known partition function p(n) stands for the number of unrestricted partitions of n, that is, the number of ways of expressing a given positive integer n as the sum of one or more positive integers, the order of the parts in the partition being irrelevant. In contrast to this, we consider here the function p*(n), which denotes the number of ways of expressing n as the product of positive integers > 2, the order of the factors in the product being irrelevant. For example, /?*(12) = 4, since 12 can be expressed in positive integers > 2 as a product in these and only these ways: 12,6 • 2, 4-3, 3-2-2. We may say that/?(n) denotes the number of sum partitions and p*(n) the number of product partitions of n. We note that the number of product partitions of n = p\axpiai.. .pkak in standard form is independent of the particular primes involved; for example, /?*(12) = p*(22 • 3) = p*(p\p2) for every choice of distinct primes p\ and/?2- For computing it is usually convenient to let pj be they'th prime; for easily ordering the divisors in increasing order it suffices to take/?2 > P\a\P?> > Pia]P2ai, etc.
    [Show full text]