DOCSLIB.ORG

Copyrighted Material

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 3 A abacus The Chinese suan pan differs from the European aba- A counting frame that started out, several thousand years cus in that the board is split into two decks, with two ago, as rows of pebbles in the desert sands of the Middle beads on each rod in the upper deck and five beads, rep- East. The word appears to come from the Hebrew âbâq resenting the digits 0 through 4, on each rod in the bot- (dust) or the Phoenician abak (sand) via the Greek abax, tom. When all five beads on a rod in the lower deck are which refers to a small tray covered with sand to hold the moved up, they’re reset to the original position, and one pebbles steady. The familiar frame-supporting rods or bead in the top deck is moved down as a carry. When wires, threaded with smoothly running beads, gradually both beads in the upper deck are moved down, they’re emerged in a variety of places and mathematical forms. reset and a bead on the adjacent rod on the left is moved In Europe, there was a strange state of affairs for more up as a carry. The result of the computation is read off than 1,500 years. The Greeks and the Romans, and then from the beads clustered near the separator beam the medieval Europeans, calculated on devices with a between the upper and lower decks. In a sense, the aba- place-value system in which zero was represented by an cus works as a 5-2-5-2-5-2...–based number system in empty line or wire. Yet the written notations didn’t have which carries and shifts are similar to those in the deci- a symbol for zero until it was introduced in Europe in mal system. Since each rod represents a digit in a deci- 1202 by Fibonacci, via the Arabs and the Hindus. mal number, the capacity of the abacus is limited only by the number of rods on the abacus. When a user runs out of rods, she simply adds another abacus to the left of the row. The Japanese soroban does away with the dual repre- sentations of fives and tens by having only four counters in the lower portion, known as “earth,” and only one counter in the upper portion, known as “heaven.” The world’s largest abacus is in the Science Museum in Lon- don and measures 4.7 meters by 2.2 meters. Abbott, Edwin Abbott (1838–1926) An English clergyman and author who wrote several the- ological works and a biography (1885) of Francis Bacon, but is best known for his standard Shakespearian Grammar (1870) and the pseudonymously written Flatland: A Romance of Many Dimensions (by A Square, 1884).[1] COPYRIGHTEDABC conjectureMATERIAL A remarkable conjecture, first put forward in 1980 by Joseph Oesterle of the University of Paris and David Masser of the Mathematics Institute of the University of Basel in Switzerland, that is now considered one of the most important unsolved problems in number theory. If it were proved correct, the proofs of many other famous conjectures and theorems would follow immediately—in some cases in just a few lines. The vastly complex current proof of Fermat’s last theorem, for example, would reduce to less than a page of mathematical reasoning. abacus A special form of the Chinese abacus (c. 1958) consist- The ABC conjecture is disarmingly simple compared ing of two abaci stacked one on top of the other. Luis Fernandes to most of the deep questions in number theory and, 3 11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 4 4 Abel, Niels Henrik moreover, turns out to be equivalent to all the main series have produced so many fallacies and so problems that involve Diophantine equations (equa- many paradoxes. tions with integer coefficients and integer solutions). A Norwegian mathematician who, independently of his Only a couple of concepts need to be understood to contemporary Évariste Galois, pioneered group theory grasp the ABC conjecture. A square-free number is an inte- and proved that there are no algebraic solutions of the ger that isn’t divisible by the square of any number. For general quintic equation. Both Abel and Galois died example, 15 and 17 are square-free, but 16 (divisible by tragically young—Abel of tuberculosis, Galois in a sword 42) and 18 (divisible by 32) are not. The square-free part of fight. an integer n, denoted sqp(n), is the largest square-free While a student in Christiania (now Oslo), Abel number that can be formed by multiplying the prime fac- thought he had discovered how to solve the general quin- tors of n. For n = 15, the prime factors are 5 and 3, and tic algebraically, but soon corrected himself in a famous 3 × 5 = 15, a square-free number, so that sqp(15) = 15. pamphlet published in 1824. In this early paper, Abel On the other hand, for n = 16, the prime factors are all 2, showed the impossibility of solving the general quintic by which means that sqp(16) = 2. In general, if n is square- means of radicals, thus laying to rest a problem that had free, the square-free part of n is just n; otherwise, sqp(n) perplexed mathematicians since the mid-sixteenth cen- represents what is left over after all the factors that create tury. Abel, chronically poor throughout his life, was a square have been eliminated. In other words, sqp(n) is granted a small stipend by the Norwegian government that the product of the distinct prime numbers that divide n. allowed him to go on a mathematical tour of Germany For example, sqp(9) = sqp(3 × 3) = 3 and sqp(1,400) = and France. In Berlin he met Leopold Crelle (1780–1856) sqp(2 × 2 × 2 × 5 × 5 × 7) = 2 × 5 × 7 = 70. and in 1826 helped him found the first journal in the The ABC conjecture deals with pairs of numbers that world devoted to mathematical research. Its first three vol- have no common factors. Suppose A and B are two such umes contained 22 of Abel’s papers, ensuring lasting fame numbers that add to give C. For example, if A = 3 and for both Abel and Crelle. Abel revolutionized the impor- B = 7, then C = 3 + 7 = 10. Now, consider the square-free tant area of elliptic integrals with his theory of elliptic part of the product A × B × C: sqp(ABC ) = sqp(3 × 7 × functions, contributed to the theory of infinite series, and 10) = 210. For most values of A and B, sqp(ABC ) > C, as founded the theory of commutative groups, known today in the prior example. In other words, sqp(ABC )/C > 1. as Abelian groups. Yet his work was never properly appre- Occasionally, however, this isn’t true. For instance, ciated during his life, and, impoverished and ill, he if A = 1 and B = 8, then C = 1 + 8 = 9, sqp(ABC ) = returned to Norway unable to obtain a teaching position. sqp(1 × 8 × 9) = sqp(1 × 2 × 2 × 2 × 3 × 3) = 1 × 2 × 3 = 6, Two days after his death, a delayed letter was delivered in and sqp(ABC )/C = 6⁄9 = 2⁄3. Similarly, if A = 3 and 15 which Abel was offered a post at the University of Berlin. B = 125, the ratio is ⁄64. David Masser proved that the ratio sqp(ABC )/C can get arbitrarily small. In other words, given any number Abelian group greater than zero, no matter how small, it’s possible to find A group that is commutative, that is, in which the result integers A and B for which sqp(ABC )/C is smaller than of multiplying one member of the group by another is this number. In contrast, the ABC conjecture says that independent of the order of multiplication. Abelian [sqp(ABC )]n/C reaches a minimum value if n is any num- groups, named after Niels Abel, are of central impor- ber greater than 1—even a number such as 1.0000000001, tance in modern mathematics, most notably in algebraic which is only barely larger than 1. The tiny change in the topology. Examples of Abelian groups include the real expression results in a huge difference in its mathematical numbers (with addition), the nonzero real numbers behavior. The ABC conjecture in effect translates an infi- (with multiplication), and all cyclic groups, such as the nite number of Diophantine equations (including the integers (with addition). equation of Fermat’s last theorem) into a single mathe- matical statement.[144] abracadabra A word famously used by magicians but which started out as a cabalistic or mystical charm for curing various Abel, Niels Henrik (1802–1829) ailments, including toothache and fever. It was first men- The divergent series are the invention of the devil, tioned in a poem called “Praecepta de Medicina” by the and it is a shame to base on them any demonstra- Gnostic physician Quintus Severus Sammonicus in the tion whatsoever. By using them, one may draw second century A.D. Sammonicus instructed that the let- any conclusion he pleases and that is why these ters be written on parchment in the form of a triangle: 11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 5 abstract algebra 5 A B R A C A D A B R A absolute zero. In some philosophies, the absolute stands A B R A C A D A B R behind the reality we see—independent, transcendent, A B R A C A D A B unconditional, and all-encompassing.
Recommended publications
  • Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory

    Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory

    Paul Erd®s and the rise of statistical thinking in elementary number theory Carl Pomerance, Dartmouth College based on the joint survey with Paul Pollack, University of Georgia 1 Let us begin at the beginning: 2 Pythagoras 3 Sum of proper divisors Let s(n) be the sum of the proper divisors of n: For example: s(10) = 1 + 2 + 5 = 8; s(11) = 1; s(12) = 1 + 2 + 3 + 4 + 6 = 16: 4 In modern notation: s(n) = σ(n) − n, where σ(n) is the sum of all of n's natural divisors. The function s(n) was considered by Pythagoras, about 2500 years ago. 5 Pythagoras: noticed that s(6) = 1 + 2 + 3 = 6 (If s(n) = n, we say n is perfect.) And he noticed that s(220) = 284; s(284) = 220: 6 If s(n) = m, s(m) = n, and m 6= n, we say n; m are an amicable pair and that they are amicable numbers. So 220 and 284 are amicable numbers. 7 In 1976, Enrico Bombieri wrote: 8 There are very many old problems in arithmetic whose interest is practically nil, e.g., the existence of odd perfect numbers, problems about the iteration of numerical functions, the existence of innitely many Fermat primes 22n + 1, etc. 9 Sir Fred Hoyle wrote in 1962 that there were two dicult astronomical problems faced by the ancients. One was a good problem, the other was not so good. 10 The good problem: Why do the planets wander through the constellations in the night sky? The not-so-good problem: Why is it that the sun and the moon are the same apparent size? 11 Perfect numbers, amicable numbers, and similar topics were important to the development of elementary number theory.
  • On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections

    On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections

    ON DYNAMICAL GASKETS GENERATED BY RATIONAL MAPS, KLEINIAN GROUPS, AND SCHWARZ REFLECTIONS RUSSELL LODGE, MIKHAIL LYUBICH, SERGEI MERENKOV, AND SABYASACHI MUKHERJEE Abstract. According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H whose limit set is an Apollonian- like gasket ΛH . We design a surgery that relates H to a rational map g whose Julia set Jg is (non-quasiconformally) homeomorphic to ΛH . We show for a large class of triangulations, however, the groups of quasisymmetries of ΛH and Jg are isomorphic and coincide with the corresponding groups of self- homeomorphisms. Moreover, in the case of H, this group is equal to the group of M¨obiussymmetries of ΛH , which is the semi-direct product of H itself and the group of M¨obiussymmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when ΛH is the classical Apollonian gasket), we give a piecewise affine model for the above actions which is quasiconformally equivalent to g and produces H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle. Contents 1. Introduction 2 2. Round Gaskets from Triangulations 4 3. Round Gasket Symmetries 6 4. Nielsen Maps Induced by Reflection Groups 12 5. Topological Surgery: From Nielsen Map to a Branched Covering 16 6. Gasket Julia Sets 18 arXiv:1912.13438v1 [math.DS] 31 Dec 2019 7.
  • He One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’S Elements?

    He One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’S Elements?

    ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT VOL. 59/2014 • ISSN 0066–6874 Zbigniew Król he One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’s Elements? ABSTRACT: his paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite diferent intuitive model (or frame), without ininite space, ininite lines and surfaces. his ancient intuitive model of Euclidean geometry is reconstructed in connection with Plato’s unwritten doctrine. he model cre- ates a kind of “hermeneutical horizon” determining the explicit content and mathematical methods used. In the irst section of the paper, it is argued that there are no actually ininite concepts in Euclid’s Elements. In the second section, it is argued that ancient mathematics is based on Plato’s highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled. KEYWORDS: Euclidean geometry • Euclid’s Elements • ancient mathematics • Plato’s unwrit- ten doctrine • philosophical hermeneutics • history of science and mathematics • philosophy of mathematics • philosophy of science he possibility to imagine all of the theorems from Euclid’s Elements in a Tquite diferent intuitive framework is interesting from the philosophical point of view. I can give one example showing how it was possible to lose the genuine image of Euclidean geometry. In many translations into Latin (Heiberg) and English (Heath) of the theorems from Book X, the term “area” (spatium) is used. For instance, the translations of X. 26 are as follows: ,,Spatium medium non excedit medium spatio rationali” and “A medial area does not exceed a medial area by a rational area”2.
  • Greece: Archimedes and Apollonius

    Greece: Archimedes and Apollonius

    Greece: Archimedes and Apollonius Chapter 4 Archimedes • “What we are told about Archimedes is a mix of a few hard facts and many legends. Hard facts – the primary sources –are the axioms of history. Unfortunately, a scarcity of fact creates a vacuum that legends happily fill, and eventually fact and legend blur into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible, to eradicate.” – Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, p. 1. Archimedes • Facts: – Lived in Syracuse – Applied mathematics to practical problems as well as more theoretical problems – Died in 212 BCE at the hands of a Roman soldier during the attack on Syracuse by the forces of general Marcellus. Plutarch, in the first century A.D., gave three different stories told about the details of his death. Archimedes • From sources written much later: – Died at the age of 75, which would put his birth at about 287 BCE (from The Book of Histories by Tzetzes, 12th century CE). – The “Eureka” story came from the Roman architect Vitruvius, about a century after Archimedes’ death. – Plutarch claimed Archimedes requested that a cylinder enclosing a sphere be put on his gravestone. Cicero claims to have found that gravestrone in about 75 CE. Archimedes • From sources written much later: – From about a century after his death come tales of his prowess as a military engineer, creating catapults and grappling hooks connected to levers that lifted boats from the sea. – Another legend has it that he invented parabolic mirrors that set ships on fire.
  • Guidance Via Number 40: a Unique Stance M

    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

    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.
  • Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna

    Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna

    Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions.
  • Large and Small Gaps Between Consecutive Niven Numbers

    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.
  • SPATIAL STATISTICS of APOLLONIAN GASKETS 1. Introduction Apollonian Gaskets, Named After the Ancient Greek Mathematician, Apollo

    SPATIAL STATISTICS of APOLLONIAN GASKETS 1. Introduction Apollonian Gaskets, Named After the Ancient Greek Mathematician, Apollo

    SPATIAL STATISTICS OF APOLLONIAN GASKETS WEIRU CHEN, MO JIAO, CALVIN KESSLER, AMITA MALIK, AND XIN ZHANG Abstract. Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the nearest neighbor spacing, pair correlation, and electrostatic energy of centers of circles from Apol- lonian gaskets. Even though the centers of these circles are not uniformly distributed in any `ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors. 1. introduction Apollonian gaskets, named after the ancient Greek mathematician, Apollonius of Perga (200 BC), are fractal sets obtained by starting from three mutually tangent circles and iter- atively inscribing new circles in the curvilinear triangular gaps. Over the last decade, there has been a resurgent interest in the study of Apollonian gaskets. Due to its rich mathematical structure, this topic has attracted attention of experts from various fields including number theory, homogeneous dynamics, group theory, and as a consequent, significant results have been obtained. Figure 1. Construction of an Apollonian gasket For example, it has been known since Soddy [23] that there exist Apollonian gaskets with all circles having integer curvatures (reciprocal of radii). This is due to the fact that the curvatures from any four mutually tangent circles satisfy a quadratic equation (see Figure 2). Inspired by [12], [10], and [7], Bourgain and Kontorovich used the circle method to prove a fascinating result that for any primitive integral (integer curvatures with gcd 1) Apollonian gasket, almost every integer in certain congruence classes modulo 24 is a curvature of some circle in the gasket.
  • Cullen Numbers with the Lehmer Property

    Cullen Numbers with the Lehmer Property

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]).
  • The Aliquot Constant We first Show the Following Result on the Geometric Mean for the Ordi- Nary Sum of Divisors Function

    The Aliquot Constant We first Show the Following Result on the Geometric Mean for the Ordi- Nary Sum of Divisors Function

    THE ALIQUOT CONSTANT WIEB BOSMA AND BEN KANE Abstract. The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proven to be less than 0. Here s(n) sums the divisors of n less than n. Thus the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded. 1. Introduction This paper is concerned with the average growth of aliquot sequences. An aliquot sequence is a sequence a0, a1, a2,... of positive integers ob- tained by iteration of the sum-of-aliquot-divisors function s, which is defined for n> 1 by s(n)= d. d|n Xd<n The aliquot sequence with starting value a0 is then equal to 2 a0, a1 = s(a0), a2 = s(a1)= s (a0),... ; k we will say that the sequence terminates (at 1) if ak = s (a0)=1 for some k ≥ 0. The sequence cycles (or is said to end in a cycle) k l 0 if s (a0) = s (a0) for some k,l with 0 ≤ l<k, where s (n) = n by arXiv:0912.3660v1 [math.NT] 18 Dec 2009 definition. Note that s is related to the ordinary sum-of-divisors function σ, with σ(n)= d|n d, by s(n)= σ(n) − n for integers n> 1.
  • Geometry and Arithmetic of Crystallographic Sphere Packings

    Geometry and Arithmetic of Crystallographic Sphere Packings

    Geometry and arithmetic of crystallographic sphere packings Alex Kontorovicha,b,1 and Kei Nakamuraa aDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08854; and bSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 21, 2018 (received for review December 12, 2017) We introduce the notion of a “crystallographic sphere packing,” argument leading to Theorem 3 comes from constructing circle defined to be one whose limit set is that of a geometrically packings “modeled on” combinatorial types of convex polyhedra, finite hyperbolic reflection group in one higher dimension. We as follows. exhibit an infinite family of conformally inequivalent crystallo- graphic packings with all radii being reciprocals of integers. We (~): Polyhedral Packings then prove a result in the opposite direction: the “superintegral” Let Π be the combinatorial type of a convex polyhedron. Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connectedz planar graph. A version of the at most, 20 dimensions. Koebe–Andreev–Thurston Theorem§ says that there exists a 3 geometrization of Π (that is, a realization of its vertices in R with sphere packings j crystallographic j arithmetic j polyhedra j straight lines as edges and faces contained in Euclidean planes) Coxeter diagrams having a midsphere (meaning, a sphere tangent to all edges). This midsphere is then also simultaneously a midsphere for the he goal of this program is to understand the basic “nature” of dual polyhedron Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket.