DOCSLIB.ORG

Table 1 Table 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Table 1 Table 2 g The ancient Egyptians expressed rational numbers as the finite sum of distinct unit fractions. Were Theorem 1. Suppose x is a positive integer and n = lcm(x, x 1,...,3, 2) + 1. Then x will have − n the Egyptians limited by this notation? In fact, they were not as every rational number can be written a length x Engel expansion. Through this project we have shown a method for choosing a denominator that will always produce as a finite sum of distinct unit fractions. Moreover, these expansions are not unique. There exist an Engel expansion whose length is equal to its numerator. We have also demonstrated that while this is not the only possible way to produce a length x expansion, n = lcm(x, x 1,...,3, 2) + 1 is several different algorithms for computing Egyptian fraction expansions, all of which produce different − conjectured to be the least denominator that does so for some fixed x. Tables 1 and 2 demonstrate representations of the same rational number. Example 2. Let x =4. Then by Theorem 1, n = lcm(4, 3, 2) + 1 = 13. In order clarify why One such algorithm, called an Engel series, produces a finite increasing sequence of integers for every this for small values of x. There are other values for n which produce a length x Engel expansion this n produces the desired expansion, 13 will be written as lcm(4, 3, 2) + 1 for the majority of this x rational number. This sequence is then used to obtain an Egyptian fraction expansion. Motivated by 4 for n. example. To find the Engel expansion, first let u1 = . The first term of the Engel series the work of M. Mays, this project aims to investigate properties of natural number denominators n lcm(4,3,2)+1 The paper includes more extensive tables for small numer- x will be 3 ators that demonstrate that n = lcm(x, x 1,...,3, 2)+1 is that produce length x Egyptian fraction expansions using Engel series for n between 0 and 1. 1 lcm(4, 3, 2) + 1 lcm(4, 3, 2) 1 Engel expansions for n − a1 = = = + . the least n that will produce an a length x Engel expansion While computing Engel expansions using Mathematica, a helpful pattern emerged: rational numbers "u # " 4 # " 4 4# x k 1 x 1 x which produce length x Egyptian fraction expansions were those whose denominator n = lcm(x, x n i=1 a1 a2 ai for . We are open to the possibility that a counter exam- n lcm(4,3,2) · ··· n − 3 1 1 x 1,...,3, 2) was divisible by every natural number between (and including) x and 2. We conjecture By the lemma, since 4 =3is an integer,a1 can be rewritten as: + ple may be found. That is, some n$ <nsuch that both n 4 ! 2 2 2 x $ · and produce x length Egyptian fraction expansions using and prove that this will always hold for length x Engel series of non-integer rational numbers between 3 1 1 n x lcm(4, 3, 2) 1 lcm(4, 3, 2) lcm(4, 3, 2) + 4 5 2 + 2 5 Engel series. None has been found yet. 0and1.Furthermore,weconjecturethatanalgorithmforfindingthesmallestn such that n produces a1 = + = +1= =4. · alengthx Engel series is to add one to the lowest common multiple of x, x 1, ...,3,and2.Aproof 4 "4# 4 4 3 1 1 1 7 3 + 3 4 + 3 4 7 − · · · that such an n works is included along with an investigation of whether or not this is the least such n. Next find u2 by subtracting one from the product of u1 and a1 to get: Table 1 In the future, we would like to determine and prove an al- 4 lcm(4, 3, 2) + 4 lcm(4, 3, 2) + 4 lcm(4, 3, 2) 1 gorithm for obtaining families of length x expansions for u2 = 1= − − x lcm(4, 3, 2) + 1 · 4 − lcm(4, 3, 2) + 1 n. That is, we are interested in defining and proving a 3 3 Engel expansions for 4 pattern that will produce each length x expansion for any = = . n lcm(4, 3, 2) + 1 13 x k 1 given x. In addition, we are interested in the ”best case” n i=1 a a a 1· 2··· i 4 1 1 1 scenarios (Engel expansions of length two). We would like The Rhind Papyrus (see Figure 1) is the best example of Egyptian 5 !2 + 2 2 + 2 2 5 The second term of the Engel series is: · · · 4 1 1 to employ methods similar to those used in this project to mathematics. It begins with expansion tables for rational numbers of 7 2 + 2 7 2 · obtain a pattern for these length 2 expansions then prove the form .ThesetablesdemonstratetheunusualwaytheEgyptians 1 lcm(4, 3, 2) + 1 lcm(4, 3, 2) 1 4 1 1 n a2 = = = + 9 3 + 3 3 that this pattern holds. · wrote out rational numbers. Instead of expressing rational numbers "u2# " 3 # " 3 3# 4 1 1 11 3 + 3 11 as proper or improper fractions, they would write them as a sum of · lcm(4,3,2) 4 1 1 1 1 1 1 2 Again, by the lemma, since =4is an integer,a can be rewritten as: 13 4 + 4 5 + 4 5 7 + 4 5 7 13 distinct unit fractions (e.g. 4 + 28 instead of 7). These expansions 3 2 · · · · · · make certain real-life problems much easier to solve. One such real- lcm(4, 3, 2) 1 lcm(4, 3, 2) lcm(4, 3, 2) + 3 Table 2 life application focuses on sharing whole items (like loaves of bread) a2 = + = +1= =5. with multiple people. For example, how can five loaves be split evenly 3 "3# 3 3 amongst eight people? The solution can be written as the proper frac- The next u term is: tion, 5.Thismeanseachpersongetsfiveeightsofaloaf(seeFigure 8 3 lcm(4, 3, 2) + 3 lcm(4, 3, 2) + 3 lcm(4, 3, 2) 1 2). This is not a very practical solution. It makes distribution of the u3 = 1= − − Figure 1: Rhind Papyrus loaves tedious (lots of slicing) and inconvenient (since no one receives lcm(4, 3, 2) + 1 · 3 − lcm(4, 3, 2) + 1 2 2 References any single larger piece of bread only small pieces). = = . lcm(4, 3, 2) + 1 13 [1] Erdos, Shallit. “New bounds on the length of finite Pierce and Engel Series.” S´eminaire de Th´eorie Luckily, this solution is not the only possible solution. Egyp- The third term in the Engel series is: des Nombres Bordeaux 3 (1991), 43-53. tian fraction expansions can be used to find a more practi- [2] Graham, Knuth, Patashink. Concrete Mathematics. Addison Wesley, Massachusetts, 1st Edition cal solution. instead of giving each person five small pieces 1 lcm(4, 3, 2) + 1 lcm(4, 3, 2) 1 (1989), Chapter 3. 5 1 1 a3 = = = + of bread, Egyptian fraction expansions show that 8 = 2 + 8. "u3# " 2 # " 2 2# [3] V. Laohakosol, T. Chaichana, J. Rattanamoong, and N.R. Kanasri. “Engel Series and Cohen-Egyptian Thus everyone will receive half a loaf first, then one eighth of lcm(4, 3, 2) 1 lcm(4, 3, 2) lcm(4, 3, 2) + 2 Fraction Expansions.” Hindawi Publishing Corporation. International Journal of Mathematics and = + = +1= =7. aloaf. ThisisnottheonlypossiblesolutionusingEgyptian 2 "2# 2 2 Mathematical Sciences. Article ID 865705 (2009), 15 pages. fractions. Another expression of the same sharing problem is [4] Mays, Michael E. “A Worst Case of the Fibonacci-Sylvester Expansion.” The Journal of Combinato- 5 1 1 1 Continue to get: rial Mathematics and Combinatorial Computing. 1 (1987), 141-148. 8 = 2 + 10 + 40. Although, this is again, an impractical solu- tion. 2 lcm(4, 3, 2) + 3 lcm(4, 3, 2) + 2 lcm(4, 3, 2) 1 [5] O’Reilly, Declan “Creating Egyptian Fractions” Mathematics in School. Vol. 21, No. 5 (Nov., 1992), u = 1= − − Page 41. 4 lcm(4, 3, 2) + 1 · 2 − lcm(4, 3, 2) + 1 [6] Renyi, A. “A New Approach to the Theory of Engel’s Series.” Annales Universitatis Scientiarum The algorithm that this poster will focus on produces an Egyptian Figure 2: Impractical Solution 1 1 = = . Budapestinensis de Rolando E¨otv¨os Nominatae. Volume 5 (1962), 25-32. fraction expansion using an Engel series. An Engel series (for rational lcm(4, 3, 2) + 1 13 numbers) is a sequence of increasing integers, a ,a ,...a [6]. The [7] Weisstein, Eric W. “Egyptian Fraction.” From MathWorld–A Wolfram Web Resource. 1 2 k http://mathworld.wolfram.com/EgyptianFraction.html sequence is infinite for irrational numbers. These integers can be used to The fourth (and hopefully last) term of the Engel series is: form an Egyptian fraction expansion by using the following summation: 1 lcm(4, 3, 2) + 1 k 1 . Any real number can be derived using an Engel series in i=1 a1 a2 ai a4 = = = lcm(4, 3, 2) + 1 = 13. auniqueway.Irrationalnumberscanberepresentedasaninfinitesum· ··· "u4# " 1 # ! [3] and rational numbers can be represented as a finite sum. An Engel Finally, series for x can be computed by defining u = x and a = 1 . From n 1 n 1 ui 1 lcm(4, 3, 2) + 1 " # u5 = 1=0 here, each subsequent ui+1 and ai+1 is obtained by first computing lcm(4, 3, 2) + 1 · 1 − Figure 3: Practical Solution 1 and the process stops. The corresponding Egyptian fraction expansion is ui+1 = ui ai 1 and ai+1 = . · − "ui+1# 4 4 1 1 1 1 I would like to thank Julia Bergner for all of her guidance.
Load more

Recommended publications
  • On Egyptian Fractions of Length 3
    ON EGYPTIAN FRACTIONS OF LENGTH 3 CYRIL BANDERIER, CARLOS ALEXIS GÓMEZ RUIZ, FLORIAN LUCA, FRANCESCO PAPPALARDI, AND ENRIQUE TREVIÑO Abstract. Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist positive a 1 1 integers m1, . , mk such that = + ··· + . Let Ak(n) be the number n m1 mk of solutions a to this equation. In this article, we give a formula for A2(p) and a parametrization for Egyptian fractions of length 3, which allows us to give a 1 1 1 bounds to A3(n), to fa(n) = #{(m1, m2, m3): = + + }, and n m1 m2 m3 a 1 1 1 finally to F (n) = #{(a, m1, m2, m3): = + + }. n m1 m2 m3 2010 Mathematics Subject Classification. 11D68, 11D45, 13A05, 01A16. Date: May 13, 2019. 1 2 C. BANDERIER, C. A. GÓMEZ RUIZ, F. LUCA, F. PAPPALARDI, AND E. TREVIÑO 1. Introduction Historical background. The most ancient mathematical texts are mostly related to computations involving proportions, fractions, inverse of integers (sometimes in link with problems related to geometry). Many traces of these mathematics are found in Sumerian or Babylonian clay tablets, during a period of several millennia1. For Egyptian mathematics, many papyri present computations involving sums of unit fractions (fractions of the form 1/n) and sometimes also the fraction 2/3; see, e.g., the Rhind Mathematical Papyrus. This document, estimated from 1550 BCE, is a copy by the scribe Ahmes of older documents. For example, it gives a list of decompositions of 2/n into unit fractions; such decompositions are also found in the Lahun Mathematical Papyri (UC 32159 and UC 32160, conserved at the University College London), which are dated circa 1800 BCE; see [27].
    [Show full text]
  • Continued Fractions and Irrationality Exponents for Modified Engel And
    Monatshefte für Mathematik https://doi.org/10.1007/s00605-018-1244-1 Continued fractions and irrationality exponents for modified Engel and Pierce series AndrewN.W.Hone1,2 · Juan Luis Varona3 Received: 29 October 2018 / Accepted: 27 November 2018 © The Author(s) 2018 Abstract An Engel series is a sum of reciprocals of a non-decreasing sequence (xn) of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property. Given an arbitrary rational number, we show that there is a family of Engel series which when added to it produces a transcendental number α whose continued fraction expansion is determined explicitly by the corresponding sequence (xn), where the latter is generated by a certain nonlinear recurrence of second order. We also present an analogous result for a rational number with a Pierce series added to or subtracted from it. In both situations (a rational number combined with either√ an Engel or a Pierce series), the irrationality exponent is bounded below by (3 + 5)/2, and we further identify infinite families of transcendental numbers α whose irrationality exponent can be computed precisely. In addition, we construct the continued fraction expansion for an arbitrary rational number added to an Engel series with the stronger property 2 that x j divides x j+1 for all j. Keywords Continued fractions · Engel series · Pierce series · Irrationality degree Communicated by A. Constantin. A. N. W. Hone: Currently on leave at School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia.
    [Show full text]
  • Pierce-Engel Hybrid Expansions
    Graduate Theses, Dissertations, and Problem Reports 2008 Pierce-Engel hybrid expansions Andrea Sutyak West Virginia University Follow this and additional works at: https://researchrepository.wvu.edu/etd Recommended Citation Sutyak, Andrea, "Pierce-Engel hybrid expansions" (2008). Graduate Theses, Dissertations, and Problem Reports. 2718. https://researchrepository.wvu.edu/etd/2718 This Dissertation is protected by copyright and/or related rights. It has been brought to you by the The Research Repository @ WVU with permission from the rights-holder(s). You are free to use this Dissertation in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you must obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Dissertation has been accepted for inclusion in WVU Graduate Theses, Dissertations, and Problem Reports collection by an authorized administrator of The Research Repository @ WVU. For more information, please contact [email protected]. Pierce-Engel Hybrid Expansions Andrea Sutyak Dissertation submitted to the Eberly College of Arts and Sciences at West Virginia University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Michael E. Mays, PhD., chair J. Goldwasser, PhD. H.W. Gould, M.A. K. Subramani, PhD. J. Wojciechowski, PhD. Department of Mathematics Morgantown, West Virginia 2008 Keywords: Pierce expansion, Engel expansion Copyright 2008 Andrea Sutyak ABSTRACT Pierce-Engel Hybrid Expansions Andrea Sutyak Pierce and Engel expansions are representations of numbers between 0 and 1 as sums of unitary fractions (of alternating signs in the case of Pierce) whose denominators are built multiplicatively, choosing the successive factors greedily.
    [Show full text]
  • Engel Expansion 1 Engel Expansion
    Engel expansion 1 Engel expansion The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If x is rational, its Engel expansion provides a representation of x as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913. An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion. Engel expansions, continued fractions, and Fibonacci Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction: They claim that ascending continued fractions such as this have been studied as early as Fibonacci's Liber Abaci (1202). This claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction: If such a notation has all numerators 0 or 1, as occurs in several instances in Liber Abaci, the result is an Engel expansion. However, Engel expansion as a general technique does not seem to be described by Fibonacci. Algorithm for computing Engel expansions To find the Engel expansion of x, let and where is the ceiling function (the smallest integer not less than r). If for any i, halt the algorithm. Engel expansion 2 Example To find the Engel expansion of 1.175, we perform the following steps. The series ends here. Thus, and the Engel expansion of 1.175 is {1, 6, 20}.
    [Show full text]
  • On Some Notes on the Engel Expansion of Ratios of Sequences Obtained from the Sum of Digits of Squared Positive Integers
    On Some Notes on the Engel Expansion of Ratios of Sequences Obtained from the Sum of Digits of Squared Positive Integers Hilary I. Okagbue*1; Abiodun A. Opanuga1; Pelumi E. Oguntunde1 and Grace Eze1,2 1Department of Mathematics, College of Science and Technology, Covenant University, Ota, Nigeria 2African Institute for Mathematical Sciences, Cameroon. *E-mail: [email protected] ABSTRACT The sequence used in this paper is the result of the digit sum of squared positive integers [5]. Objective: To introduce a new novel approach to Similar works done on the sum and iterative digit the understanding of Engel expansions of ratios of sums of selected integer sequences can be number sequences. found in the cases of cubed positive integers [6], Sophie Germain and Safe primes [7], Methodology: Let C be the ratios of consecutive Palindromic, Repdigit and Repunit numbers [8], elements of sequence obtained from the sum of Fibonacci numbers [9] and so on. digits of squared positive integers and D be the ratios of consecutive elements of sequence that is The sequence generated by the sum of digit of the complement of C but the domain is the squared positive integers [5] is given as: positive integer. Engel series expansions and Pierce expansion were obtained for C while only 1, 4, 7, 9, 10, 13, 16, 18, 19, ... (A) Engel series expansions were obtained for D. And the sequence obtained from the complement Findings: The distance between the respective of (A) is: Engel series and their means for the two sequences are neither unique nor uniformly distributed.
    [Show full text]
  • Continued Fractions and Irrationality Exponents for Modified Engel and Pierce Series
    Kent Academic Repository Full text document (pdf) Citation for published version Hone, Andrew N.W. and Varona, Juan Luis (2018) Continued fractions and irrationality exponents for modified engel and pierce series. Monatshefte fur Mathematik . pp. 1-16. ISSN 0026-9255. DOI https://doi.org/10.1007/s00605-018-1244-1 Link to record in KAR https://kar.kent.ac.uk/70203/ Document Version Publisher pdf Copyright & reuse Content in the Kent Academic Repository is made available for research purposes. Unless otherwise stated all content is protected by copyright and in the absence of an open licence (eg Creative Commons), permissions for further reuse of content should be sought from the publisher, author or other copyright holder. Versions of research The version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record. Enquiries For any further enquiries regarding the licence status of this document, please contact: [email protected] If you believe this document infringes copyright then please contact the KAR admin team with the take-down information provided at http://kar.kent.ac.uk/contact.html Monatshefte für Mathematik https://doi.org/10.1007/s00605-018-1244-1 Continued fractions and irrationality exponents for modified Engel and Pierce series AndrewN.W.Hone1,2 · Juan Luis Varona3 Received: 29 October 2018 / Accepted: 27 November 2018 © The Author(s) 2018 Abstract An Engel series is a sum of reciprocals of a non-decreasing sequence (xn) of positive integers, which is such that each term is divisible by the previous one, and a Pierce series is an alternating sum of the reciprocals of a sequence with the same property.
    [Show full text]
  • Engel Expansion 1 Engel Expansion
    Engel expansion 1 Engel expansion The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that Rational numbers have a finite Engel expansion, while irrational numbers have an infinite Engel expansion. If x is rational, its Engel expansion provides a representation of x as an Egyptian fraction. Engel expansions are named after Friedrich Engel, who studied them in 1913. An expansion analogous to an Engel expansion, in which alternating terms are negative, is called a Pierce expansion. Engel expansions, continued fractions, and Fibonacci Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction: They claim that ascending continued fractions such as this have been studied as early as Fibonacci's Liber Abaci (1202). This claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction: If such a notation has all numerators 0 or 1, as occurs in several instances in Liber Abaci, the result is an Engel expansion. However, Engel expansion as a general technique does not seem to be described by Fibonacci. Algorithm for computing Engel expansions To find the Engel expansion of x, let and where is the ceiling function (the smallest integer not less than r). If for any i, halt the algorithm. Engel expansion 2 Example To find the Engel expansion of 1.175, we perform the following steps. The series ends here. Thus, and the Engel expansion of 1.175 is {1, 6, 20}.
    [Show full text]
  • Tutorme Subjects Covered.Xlsx
    Subject Group Subject Topic Computer Science Android Programming Computer Science Arduino Programming Computer Science Artificial Intelligence Computer Science Assembly Language Computer Science Computer Certification and Training Computer Science Computer Graphics Computer Science Computer Networking Computer Science Computer Science Address Spaces Computer Science Computer Science Ajax Computer Science Computer Science Algorithms Computer Science Computer Science Algorithms for Searching the Web Computer Science Computer Science Allocators Computer Science Computer Science AP Computer Science A Computer Science Computer Science Application Development Computer Science Computer Science Applied Computer Science Computer Science Computer Science Array Algorithms Computer Science Computer Science ArrayLists Computer Science Computer Science Arrays Computer Science Computer Science Artificial Neural Networks Computer Science Computer Science Assembly Code Computer Science Computer Science Balanced Trees Computer Science Computer Science Binary Search Trees Computer Science Computer Science Breakout Computer Science Computer Science BufferedReader Computer Science Computer Science Caches Computer Science Computer Science C Generics Computer Science Computer Science Character Methods Computer Science Computer Science Code Optimization Computer Science Computer Science Computer Architecture Computer Science Computer Science Computer Engineering Computer Science Computer Science Computer Systems Computer Science Computer Science Congestion Control
    [Show full text]
  • Generalized Expansions of Real Numbers
    Generalized expansions of real numbers by Simon Plouffe First Draft, April 2006 Revised August 28, 2014 Abstract I present here a collection of algorithms that permits the expansion into a finite series or sequence from a real number ∈ , the precision used is 64 decimal digits. The collection of mathematical constants was taken from my own collection and theses sources [1]‐[6][9][10]. The goal of this experiment is to find a closed form of the sequence generated by the algorithm. Some new results are presented. ‐Introduction ‐Algorithms ‐Results ‐Appendix ‐Introduction Most of the algorithms will produce a sequence of integers when ∈ and can be written as a 2 terms recurrence. If is the initial value then will be the terms of the sequence. If is rational then most of the algorithms will lead to a finite sequence. But with 64 decimal digits enough terms are computed for detecting simple patterns as with some quadratic irrational like √2 or the Golden Ration. Other numbers like , : 2.71828… do have a pattern which is easily recognizable but most real numbers do not. The goal of this computation of sequences from real numbers using different algorithms is to discover or find if there could be any patterns at all with other algorithms. The natural question that comes to mind is: is there any closed formula or generating function for those sequences? For this I can use Gfun package of Maple or with Mathematica as well to answer the question. Gfun was developed starting in 1991 by me and François Bergeron, see [7].
    [Show full text]
  • Some Properties of Numbers and Functions in Analytic Number Theory
    Some properties of numbers and functions in analytic number theory Soichi Ikeda 1 2 Contents 1. Introduction 3 1.1. Zeta-functionsandarithmeticfunctions 3 1.2. Irrational numbers and Transcendental numbers 5 1.3. Summary of Section 2 6 1.4. Summary of Section 3 7 1.5. Summary of Section 4 7 2. DoubleanalogueofHamburger’stheorem 8 2.1. Introduction 8 2.2. LemmasfortheproofofTheorem2.1 11 2.3. Proof of Theorem 2.1 13 3. Onthelcm-sumfunction 16 3.1. Introduction 16 3.2. Lemmasfortheproofoftheorems 18 3.3. ProofofTheorem3.1andTheorem3.2 20 3.4. Proof of Theorem 3.3 21 4. A new construction of the real numbers by alternating series 25 4.1. Introduction 25 4.2. Fundamental properties of the generalized alternating Sylvester series 28 4.3. Constructionoftherealnumbers 32 4.4. An application 40 References 41 3 1. Introduction In this paper, we discuss some properties of real numbers and some functions in analytic number theory. This paper is divided into the following four sections. 1. Introduction 2. Double analogue of Hamburger’s theorem 3. On the lcm-sum function 4. A new construction of the real numbers by alternating series Sections 2, 3 and 4 are devoted to the explanation of our main results. Section 1 (this section) is divided into the following five parts. 1.1. Zeta-functions and arithmetic functions 1.2. Irrational numbers and Transcendental numbers 1.3. Summary of Section 2 1.4. Summary of Section 3 1.5. Summary of Section 4 Since the theory of zeta-functions and arithmetic functions and the theory of irrational numbers and transcendental numbers are important in analytic number theory, we show some facts in these theories in Sections 1.1 and 1.2.
    [Show full text]
  • On Egyptian Fractions of Length 3
    REVISTA DE LA UNION´ MATEMATICA´ ARGENTINA Vol. 62, No. 1, 2021, Pages 257–274 Published online: June 30, 2021 https://doi.org/10.33044/revuma.1798 ON EGYPTIAN FRACTIONS OF LENGTH 3 CYRIL BANDERIER, CARLOS ALEXIS GOMEZ´ RUIZ, FLORIAN LUCA, FRANCESCO PAPPALARDI, AND ENRIQUE TREVINO˜ Abstract. Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist a 1 1 positive integers m1, . , mk such that = + ··· + . Let Ak(n) be the n m1 mk number of solutions a to this equation. In this article, we give a formula for A2(p) and a parametrization for Egyptian fractions of length 3, which allows us a 1 1 1 to give bounds to A3(n), to fa(n) = #{(m1, m2, m3): = + + }, n m1 m2 m3 a 1 1 1 and finally to F (n) = #{(a, m1, m2, m3): = + + }. n m1 m2 m3 1. Introduction Historical background. The most ancient mathematical texts are mostly re- lated to computations involving proportions, fractions, and inverses of integers (sometimes connected with problems related to geometry). Many traces of these mathematics are found in Sumerian or Babylonian clay tablets covering a period of several millennia1. For Egyptian mathematics, many papyri present computations involving sums of unit fractions (fractions of the form 1/n) and sometimes also the fraction 2/3; see e.g. the Rhind Mathematical Papyrus. This document, estimated from 1550 BCE, 2020 Mathematics Subject Classification. 11D68, 11D45, 13A05, 01A16. Key words and phrases. Diophantine equations; Egyptian fractions; divisibility; history of mathematics; Erd˝os–Strausconjecture; asymptotic estimate.
    [Show full text]
  • Arxiv:1707.06139V2 [Math.HO] 15 Jun 2018 the Sequence of Approximants
    A CHRONOLOGY OF CONTINUED SQUARE ROOTS AND OTHER CONTINUED COMPOSITIONS, THROUGH THE YEAR 2016 DIXON J. JONES Abstract. An infinite continued composition is an expression of the form lim t0 ◦ t1 ◦ t2 ◦ · · · ◦ tn(c) ; n!1 where the ti are maps from a set D to itself, the initial value c is a point in D, and the order of operations proceeds from right to left. This document is an annotated bibliography, in chronological order through the year 2016, of selected continued compositions whose primary sources have typically been obscure. In particular, we include continued square roots: r q p a0 + a1 + a2 + :::; as well as continued powers, continued cotangents, and f-expansions. However, we do not include continued fractions, continued exponentials, or forms such as infinite sums and products in which the ti are linear functions, because the literature on these forms is extensive and well-summarized elsewhere. 1. Introduction This is a bibliography, in chronological order through the year 2016, of selected continued function compositions or, more briefly, continued compositions 1, expres- sions of the form lim t0 ◦ t1 ◦ t2 ◦ · · · ◦ tn(c) ; (1) n!1 where the ti are maps from a set D to itself, the initial value c is a point in D, and the order of operations proceeds from right to left. Under these conditions the nth approximant t0 ◦ t1 ◦ t2 ◦ · · · ◦ tn(c) (2) is well-defined for each n = 0; 1; 2;:::, and the continued composition converges if arXiv:1707.06139v2 [math.HO] 15 Jun 2018 the sequence of approximants t0(c); t0 ◦ t1(c); t0 ◦ t1 ◦ t2(c); (3) Date: June 19, 2018.
    [Show full text]