DOCSLIB.ORG

ON FINDING the SMALLEST HAPPY NUMBERS of ANY HEIGHTS 2 Where B(0) = 0 in Base B

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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). We note x as a vector of its digits in base B as being x = (xL(x)−1,...,x0)B where we note L(x) as the length of x and x(k) the kth digit of x starting from the left for k 0, 1,...,L(x) 1 . ∈{ − } Definition 1.1. Let B 2 be the numerical base of a positive integer x. We define ≥ a function HB : N N where −→ L(x)−1 2 B(x)= x , (2) H i i=0 X 2010 Mathematics Subject Classification. Mathematics Subject Classification 2010: 11B37, 11Y55, 11Y16, 11A07. Key words and phrases. Number theory, Smallest happy numbers, Height, Algorithm, Modular arithmetic. 1 ON FINDING THE SMALLEST HAPPY NUMBERS OF ANY HEIGHTS 2 where B(0) = 0 in base B. H n n−1 To simplify the notation, we note B (x)= B (x) B (x) for all n 1, where is the function composition operator.H The identityH is◦H defined as 0 (x≥)= x. ◦ HB Definition 1.2. Let B 2 be the numerical base of a positive integer x. We call x a happy number if and≥ only if there is n N such that n (x)=1. ∈ HB Definition 1.3. We define the height of a happy number x in base B 2 as ≥ ηB : N∗ N where −→ α ηB (x) = min α N : (x)=1 . (3) { ∈ HB } Definition 1.4. We define the smallest happy number in numerical base B 2 ≥ for any height by the function γB : N N∗ where −→ η(x) γB(η(x)) = min x N∗ : (x)=1 . (4) { ∈ HB } To simplify the notation, we will use η = ηB(x) and γ(η) = γB(η). If there is ambiguity, we will precise the numerical base (e.g. γ3(η3)). 2. Smallest happy numbers The objective of this section is to show that there is a relationship between subsequent heights of smallest happy numbers for all base B 2. Then, we get a general equation for γ on which the algorithm will be based. ≥ The following result shows that the function γ is defined uniquely for every η. Theorem 2.1. For all η N, there is a unique x N∗ such that γ(η) = x and ∈ ∈ B(γ(η + 1)) γ(η). H ≥ Proof. To prove the existence, we have to show that there is a x N∗ happy such that η(x) = y. We proceed by induction on y. For y = 0, we have∈ x = 1 such that η(x) = 0 for all B 2. Lets assume that there is a x N∗ happy such that ≥ ∈ η(x) = y. Let z = (1,..., 1)B such that L(z) = x. Thus, B(z) = x and z is also happy. In virtue of the induction hypothesis, η(z) = y +H 1 which proves the existence for all y N. ∈ Let x , x N∗ where x = x . Then, let x > x without loss of generality. 1 2 ∈ 1 6 2 1 2 Having γ(η)= x1 or γ(η)= x2 contradict the definition 1.4. Thus, x1 = x2 which proves the uniqueness. We deduce from the definition 1.4 that B(γ(η + 1)) γ(η). H ≥ We note that 1 is the unique integer of height 0. If 1 <x<B, then B(x) = 2 H x > x. This ensures that (1, 0)B is the smallest happy number of height 1 for all B 2. ≥ Lemma 2.2. If γ(η + 1) B2, then γ(η + 1) >γ(η). ≥ Proof. By theorem 2.1, γ(η + 1) = γ(η) for all η. We proceed by contrapositive and suppose that γ(η) > γ(η + 1).6 Since γ(η + 1) B2, we have, in virtue of the ≥ lemma 6 [1], that B(γ(η + 1)) < γ(η + 1). Applying this with our hypothesis H implies that B(γ(η + 1)) <γ(η) a contradiction with the theorem 2.1. Therefore, γ(η + 1) >γH(η). ON FINDING THE SMALLEST HAPPY NUMBERS OF ANY HEIGHTS 3 We want to show that the smallest happy numbers are strictly increasing as their height increases starting at a certain height when γ(η + 1) > 2(B 1)2. However, even if B2 γ(η + 1) 2(B 1)2, it is generally not true. For− example, using ≤ ≤ − 2 brute-force, we calculate that γ18(28) = (1, 2, 17)18 >B and γ18(29) = (11, 16)18. We see that γ18(28) >γ18(29). The next result ensure that γ(η),γ(η + 1),... is strictly increasing when γ(η + 1) > 2(B 1)2 for all B 3 and{ γ(η + 1) B2}for the binary base. − ≥ ≥ Lemma 2.3. If γ(η + 1) > 2(B 1)2 for all B 3 and γ(η + 1) B2 if B =2, then γ(η) <γ(η + 1) <...<γ(η−+ k) for all k ≥N. ≥ ∈ Proof. Let’s proceed by induction on k. For the base case, we have to show that γ(η) <γ(η + 1). Since γ(η + 1) > 2(B 1)2 B2 for all B 3 and γ(η + 1) B2 if B = 2, we apply the lemma 2.2 and− then≥ get the result.≥ Suppose that γ(≥η) < γ(η + 1) <...<γ(η + k) holds for a certain k. We have to show that it also holds for k + 1. Suppose that γ(η+k+1) 2(B 1)2 for all B 3 and γ(η+k+1) <B2 if B = 2. ≤ − N ≥ n 2 In virtue of the corollary 7 [1], there is n such that B(γ(η + k + 1)) < B . n+1 ∈ 2 H ′ It follows that B (γ(η + k + 1)) 2(B 1) which is maximal if any γ(η ) = H ≤ −′ (B 1,B 1)B for all n < η + k + 1 and η η + k + 1. By induction on n with the− theorem− 2.1, one shows that γ(η′) 2(B ≤1)2 for all η′ η + k +1. Therefore, it is generally false to say that γ(0) <γ≤(1) <...<γ− (η′). ≤ Suppose instead that γ(η+k+1) > 2(B 1)2 for all B 3 and γ(η+k+1) B2 if B = 2. By the lemma 2.2, we have γ(η +−k + 1) >γ(η +≥k). Using the induction≥ hypothesis, we get γ(η) < γ(η + 1) <...<γ(η + k) < γ(η + k + 1). Therefore, γ(η) <γ(η + 1) <...<γ(η + k) for all k N. ∈ Let [x] be the integer part function of x and x be the ceiling function of x. The operator in the expression a b denotes the⌈ concatenation⌉ of a and b. ∗ ∗ Let Lη = αη + tη denotes the total number of digits of γ(η), where αη is the number of digits lower than B 1 and tη the number of B 1. We also note − − γ(η)= Aη Tη where Aη is the integer containing the digits lower than B 1 and ∗ − Tη the digits B 1. The next results give boundaries on αη and tη. − γ(η) Corollary 2.4. We have tη 2 (2B 4) for all B 2. +1 ≥ (B−1) − − ≥ h i γ(η) Proof. Let yη+1 = (B−1)2 . By definition of tη+1, we know that h i αη = Lη tη . (5) +1 +1 − +1 In virtue of the lemma 2.1 [4], yη Lη yη + 4. With the equation (5), +1 ≤ +1 ≤ +1 it implies that αη+1 yη+1 +4 tη+1 and yη+1 αη+1 + tη+1. With γ(η) = 2 ≤ − ≤ (B 1) yη = Aη Tη, we deduce that − +1 ∗ 2 2 2 (B 1) yη (B 2) (yη +4 tη ) + (B 1) tη . (6) − +1 ≤ − +1 − +1 − +1 The inequation (6) is equivalent to 2 (2B 3)yη (4B 16B +16)+(2B 3)tη − +1 ≤ − − +1 4B2−16B+16 1 which gives tη yη = yη 2B +5 .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Grade 7/8 Math Circles the Scale of Numbers Introduction
    Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero.
    [Show full text]
  • Power Towers and the Tetration of Numbers
    POWER TOWERS AND THE TETRATION OF NUMBERS Several years ago while constructing our newly found hexagonal integer spiral graph for prime numbers we came across the sequence- i S {i,i i ,i i ,...}, Plotting the points of this converging sequence out to the infinite term produces the interesting three prong spiral converging toward a single point in the complex plane as shown in the following graph- An inspection of the terms indicate that they are generated by the iteration- z[n 1] i z[n] subject to z[0] 0 As n gets very large we have z[]=Z=+i, where =exp(-/2)cos(/2) and =exp(-/2)sin(/2). Solving we find – Z=z[]=0.4382829366 + i 0.3605924718 It is the purpose of the present article to generalize the above result to any complex number z=a+ib by looking at the general iterative form- z[n+1]=(a+ib)z[n] subject to z[0]=1 Here N=a+ib with a and b being real numbers which are not necessarily integers. Such an iteration represents essentially a tetration of the number N. That is, its value up through the nth iteration, produces the power tower- Z Z n Z Z Z with n-1 zs in the exponents Thus- 22 4 2 22 216 65536 Note that the evaluation of the powers is from the top down and so is not equivalent to the bottom up operation 44=256. Also it is clear that the sequence {1 2,2 2,3 2,4 2,...}diverges very rapidly unlike the earlier case {1i,2 i,3i,4i,...} which clearly converges.
    [Show full text]
  • Attributes of Infinity
    International Journal of Applied Physics and Mathematics Attributes of Infinity Kiamran Radjabli* Utilicast, La Jolla, California, USA. * Corresponding author. Email: [email protected] Manuscript submitted May 15, 2016; accepted October 14, 2016. doi: 10.17706/ijapm.2017.7.1.42-48 Abstract: The concept of infinity is analyzed with an objective to establish different infinity levels. It is proposed to distinguish layers of infinity using the diverging functions and series, which transform finite numbers to infinite domain. Hyper-operations of iterated exponentiation establish major orders of infinity. It is proposed to characterize the infinity by three attributes: order, class, and analytic value. In the first order of infinity, the infinity class is assessed based on the “analytic convergence” of the Riemann zeta function. Arithmetic operations in infinity are introduced and the results of the operations are associated with the infinity attributes. Key words: Infinity, class, order, surreal numbers, divergence, zeta function, hyperpower function, tetration, pentation. 1. Introduction Traditionally, the abstract concept of infinity has been used to generically designate any extremely large result that cannot be measured or determined. However, modern mathematics attempts to introduce new concepts to address the properties of infinite numbers and operations with infinities. The system of hyperreal numbers [1], [2] is one of the approaches to define infinite and infinitesimal quantities. The hyperreals (a.k.a. nonstandard reals) *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + … + 1, which is infinite number, and its reciprocal is infinitesimal. Also, the set theory expands the concept of infinity with introduction of various orders of infinity using ordinal numbers.
    [Show full text]
  • 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.
    [Show full text]
  • The Notion Of" Unimaginable Numbers" in Computational Number Theory
    Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers.
    [Show full text]
  • 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.
    [Show full text]
  • Hyperoperations and Nopt Structures
    Hyperoperations and Nopt Structures Alister Wilson Abstract (Beta version) The concept of formal power towers by analogy to formal power series is introduced. Bracketing patterns for combining hyperoperations are pictured. Nopt structures are introduced by reference to Nept structures. Briefly speaking, Nept structures are a notation that help picturing the seed(m)-Ackermann number sequence by reference to exponential function and multitudinous nestings thereof. A systematic structure is observed and described. Keywords: Large numbers, formal power towers, Nopt structures. 1 Contents i Acknowledgements 3 ii List of Figures and Tables 3 I Introduction 4 II Philosophical Considerations 5 III Bracketing patterns and hyperoperations 8 3.1 Some Examples 8 3.2 Top-down versus bottom-up 9 3.3 Bracketing patterns and binary operations 10 3.4 Bracketing patterns with exponentiation and tetration 12 3.5 Bracketing and 4 consecutive hyperoperations 15 3.6 A quick look at the start of the Grzegorczyk hierarchy 17 3.7 Reconsidering top-down and bottom-up 18 IV Nopt Structures 20 4.1 Introduction to Nept and Nopt structures 20 4.2 Defining Nopts from Nepts 21 4.3 Seed Values: “n” and “theta ) n” 24 4.4 A method for generating Nopt structures 25 4.5 Magnitude inequalities inside Nopt structures 32 V Applying Nopt Structures 33 5.1 The gi-sequence and g-subscript towers 33 5.2 Nopt structures and Conway chained arrows 35 VI Glossary 39 VII Further Reading and Weblinks 42 2 i Acknowledgements I’d like to express my gratitude to Wikipedia for supplying an enormous range of high quality mathematics articles.
    [Show full text]
  • 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”.
    [Show full text]
  • Mathematical Constants and Sequences
    Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan's Library, ISSN 2421-1230, Vol.II. First release March 31, 2008. Permalink via DOI: 10.3247/SL2Math08.001 This page is dedicated to my late math teacher Jaroslav Bayer who, back in 1955-8, kindled my passion for Mathematics. Math BOOKS | SI Units | SI Dimensions PHYSICS Constants (on a separate page) Mathematics LINKS | Stan's Library | Stan's HUB This is a constant-at-a-glance list. You can also download a PDF version for off-line use. But keep coming back, the list is growing! When a value is followed by #t, it should be a proven transcendental number (but I only did my best to find out, which need not suffice). Bold dots after a value are a link to the ••• OEIS ••• database. This website does not use any cookies, nor does it collect any information about its visitors (not even anonymous statistics). However, we decline any legal liability for typos, editing errors, and for the content of linked-to external web pages. Basic math constants Binary sequences Constants of number-theory functions More constants useful in Sciences Derived from the basic ones Combinatorial numbers, including Riemann zeta ζ(s) Planck's radiation law ... from 0 and 1 Binomial coefficients Dirichlet eta η(s) Functions sinc(z) and hsinc(z) ... from i Lah numbers Dedekind eta η(τ) Functions sinc(n,x) ... from 1 and i Stirling numbers Constants related to functions in C Ideal gas statistics ... from π Enumerations on sets Exponential exp Peak functions (spectral) ..
    [Show full text]
  • Reihenalgebra: What Comes Beyond Exponentiation? M
    Reihenalgebra: What comes beyond exponentiation? M. M¨uller, [email protected] Abstract Addition, multiplication and exponentiation are classical oper- ations, successively defined by iteration. Continuing the iteration process, one gets an infinite hierarchy of higher-order operations, the first one sometimes called tetration a... aa a b = a b terms , ↑ o followed by pentation, hexation, etc. This paper gives a survey on some ideas, definitions and methods that the author has developed as a young pupil for the German Jugend forscht science fair. It is meant to be a collection of ideas, rather than a serious formal paper. In particular, a definition for negative integer exponents b is given for all higher-order operations, and a method is proposed that gives a very natural (but non-trivial) interpolation to real (and even complex) integers b for pentation and hexation and many other operations. It is an open question if this method can also be applied to tetration. 1 Introduction Multiplication of natural numbers is nothing but repeated addition, a + a + a + ... + a = a b . (1) · b terms | {z } Iterating multiplication, one gets another operation, namely exponentiation: b a a a ... a = a =: aˆb . (2) · · · · b terms | {z } Classically, this is it, and one stops here. But what if one continues the iteration process? One could define something like aˆaˆaˆ ... ˆa = a b . ↑ b terms | {z } But, wait a minute, in contrast to eq. (1) and (6), this definition will depend on the way we set brackets, i.e. on the order of exponentiation! Thus, we have to distinguish between the two canonical possibilities aaa..
    [Show full text]
  • The Strange Properties of the Infinite Power Tower Arxiv:1908.05559V1
    The strange properties of the infinite power tower An \investigative math" approach for young students Luca Moroni∗ (August 2019) Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. Paul Lockhart { \A Mathematician's Lament" Abstract In this article we investigate some "unexpected" properties of the \Infinite Power Tower 1" function (or \Tetration with infinite height"): . .. xx y = f(x) = xx where the \tower" of exponentiations has an infinite height. Apart from following an initial personal curiosity, the material collected here is also intended as a potential guide for teachers of high-school/undergraduate students interested in planning an activity of \investigative mathematics in the classroom", where the knowledge is gained through the active, creative and cooperative use of diversified mathematical tools (and some ingenuity). The activity should possibly be carried on with a laboratorial style, with no preclusions on the paths chosen and undertaken by the students and with little or no information imparted from the teacher's desk. The teacher should then act just as a guide and a facilitator. The infinite power tower proves to be particularly well suited to this kind of learning activity, as the student will have to face a challenging function defined through a rather uncommon infinite recursive process. They'll then have to find the right strategies to get around the trickiness of this function and achieve some concrete results, without the help of pre-defined procedures. The mathematical requisites to follow this path are: functions, properties of exponentials and logarithms, sequences, limits and derivatives.
    [Show full text]