Music of the Primes in Search of Order
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Ces`Aro's Integral Formula for the Bell Numbers (Corrected)
Ces`aro’s Integral Formula for the Bell Numbers (Corrected) DAVID CALLAN Department of Statistics University of Wisconsin-Madison Medical Science Center 1300 University Ave Madison, WI 53706-1532 [email protected] October 3, 2005 In 1885, Ces`aro [1] gave the remarkable formula π 2 cos θ N = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin pθ dθ p πe Z0 where (Np)p≥1 = (1, 2, 5, 15, 52, 203,...) are the modern-day Bell numbers. This formula was reproduced verbatim in the Editorial Comment on a 1941 Monthly problem [2] (the notation Np for Bell number was still in use then). I have not seen it in recent works and, while it’s not very profound, I think it deserves to be better known. Unfortunately, it contains a typographical error: a factor of p! is omitted. The correct formula, with n in place of p and using Bn for Bell number, is π 2 n! cos θ B = ee cos(sin θ)) sin( ecos θ sin(sin θ) ) sin nθ dθ n ≥ 1. n πe Z0 eiθ The integrand is the imaginary part of ee sin nθ, and so an equivalent formula is π 2 n! eiθ B = Im ee sin nθ dθ . (1) n πe Z0 The formula (1) is quite simple to prove modulo a few standard facts about set par- n titions. Recall that the Stirling partition number k is the number of partitions of n n [n] = {1, 2,...,n} into k nonempty blocks and the Bell number Bn = k=1 k counts n k n n k k all partitions of [ ]. -
Squaring, Cubing, and Cube Rooting Arthur T
Squaring, Cubing, and Cube Rooting Arthur T. Benjamin Arthur T. Benjamin ([email protected]) has taught at Harvey Mudd College since 1989, after earning his Ph.D. from Johns Hopkins in Mathematical Sciences. He has served MAA as past editor of Math Horizons and the Spectrum Book Series, has written two books for MAA, and served as Polya Lecturer from 2006 to 2008. He frequently performs as a mathemagician, a term popularized by Martin Gardner. I still recall the thrill and simultaneous disappointment I felt when I first read Math- ematical Carnival [4] by Martin Gardner. I was thrilled because, as my high school teacher had told me, mathematics was presented there in a playful way that I had never seen before. I was disappointed because Gardner quoted a formula that I thought I had “invented” a few years earlier. I have always had a passion for mental calculation, and the formula (1) below appears in Gardner’s chapter on “Lightning Calculators.” It was used by the mathematician A. C. Aitken to square large numbers mentally. Squaring Aitken took advantage of the following algebraic identity. A2 (A d)(A d) d2. (1) = − + + Naturally, this formula works for any value of d, but we should choose d to be the distance to a number close to A that is easy to multiply. Examples. To square the number 23, we let d 3 to get = 232 20 26 32 520 9 529. = × + = + = To square 48, let d 2 to get = 482 50 46 22 2300 4 2304. = × + = + = With just a little practice, it’s possible to square any two-digit number in a matter of seconds. -
A Look at the ABC Conjecture Via Elliptic Curves
A Look at the ABC Conjecture via Elliptic Curves Nicole Cleary Brittany DiPietro Alexander Hill Gerard D.Koffi Beihua Yan Abstract We study the connection between elliptic curves and ABC triples. Two important results are proved. The first gives a method for finding new ABC triples. The second result states conditions under which the power of the new ABC triple increases or decreases. Finally, we present two algorithms stemming from these two results. 1 Introduction The ABC conjecture is a central open problem in number theory. It was formu- lated in 1985 by Joseph Oesterl´eand David Masser, who worked separately but eventually proposed equivalent conjectures. Like many other problems in number theory, the ABC conjecture can be stated in relatively simple, understandable terms. However, there are several profound implications of the ABC conjecture. Fermat's Last Theorem is one such implication which we will explore in the second section of this paper. 1.1 Statement of The ABC Conjecture Before stating the ABC conjecture, we define a few terms that are used frequently throughout this paper. Definition 1.1.1 The radical of a positive integer n, denoted rad(n), is defined as the product of the distinct prime factors of n. That is, rad(n) = Q p where p is prime. pjn 1 Example 1.1.2 Let n = 72 = 23 · 32: Then rad(n) = rad(72) = rad(23 · 32) = 2 · 3 = 6: Definition 1.1.3 Let A; B; C 2 Z. A triple (A; B; C) is called an ABC triple if A + B = C and gcd(A; B; C) = 1. -
On Cullen Numbers Which Are Both Riesel and Sierpiński Numbers
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Number Theory 132 (2012) 2836–2841 Contents lists available at SciVerse ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt On Cullen numbers which are both Riesel and Sierpinski´ numbers ∗ Pedro Berrizbeitia a, J.G. Fernandes a, ,MarcosJ.Gonzáleza,FlorianLucab, V. Janitzio Mejía Huguet c a Departamento de Matemáticas, Universidad Simón Bolívar, Caracas 1080-A, Venezuela b Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089, Morelia, Michoacán, Mexico c Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco, Av. San Pablo #180, Col. Reynosa Tamaulipas, Azcapotzalco, 02200, México DF, Mexico article info abstract Article history: We describe an algorithm to determine whether or not a given Received 27 January 2012 system of congruences is satisfied by Cullen numbers. We use Revised 21 May 2012 this algorithm to prove that there are infinitely many Cullen Accepted 21 May 2012 numbers which are both Riesel and Sierpinski.´ (Such numbers Available online 3 August 2012 should be discarded if you are searching prime numbers with Communicated by David Goss Proth’s theorem.) Keywords: © 2012 Elsevier Inc. All rights reserved. Cullen numbers Covering congruences Sierpinski´ numbers Riesel numbers 1. Introduction n The numbers Cn = n2 + 1 where introduced, in 1905, by the Reverend J. Cullen [1] as a particular n n case of the Proth family k2 + 1, where k = n. The analogous numbers Wn = n2 − 1 where intro- duced as far as we know, in 1917, by Cunningham and Woodall [3]. -
Divisibility Rules by James D
DIVISIBILITY RULES BY JAMES D. NICKEL Up to this point in our study, we have explored addition, subtraction, and multiplication of whole num- bers. Recall that, on average, out of every 100 arithmetical problems you encounter 70 will require addition, 20 will require multiplication, 5 will require subtraction, and 5 will require division. Although least used, di- vision is a part of everyday life. Parents, in providing inheritances for their children, generally seek to divide the estate equally between their children. When four people play with a deck of 52 cards, they seek to divide the cards equally to each player. We are now ready to explore the nature of division and, to speak truthfully, division is one of the hard- est processes for a student to master because it is the most complicated of all the arithmetical operations. Note what was said about it in 1600: Division is esteemed one of the busiest operations of Arithmetick, and such as requireth a mynde not wandering, or setled uppon other matters.1 So, we shall tread this division ground lightly before we start digging. We shall provide a few lessons of “interlude” first that I think you will find very enjoyable. The ancient Greeks of the Classical era (ca. 600–300 BC) were fascinated with the structure of number, especially the natural numbers. For the = 2 = 1 philosophers of this period, the study of the structure of natural numbers = 4 = 3 was a favorite “thought” hobby. We have already seen one example of this in the classification of even and odd numbers. -
MAT344 Lecture 6
MAT344 Lecture 6 2019/May/22 1 Announcements 2 This week This week, we are talking about 1. Recursion 2. Induction 3 Recap Last time we talked about 1. Recursion 4 Fibonacci numbers The famous Fibonacci sequence starts like this: 1; 1; 2; 3; 5; 8; 13;::: The rule defining the sequence is F1 = 1;F2 = 1, and for n ≥ 3, Fn = Fn−1 + Fn−2: This is a recursive formula. As you might expect, if certain kinds of numbers have a name, they answer many counting problems. Exercise 4.1 (Example 3.2 in [KT17]). Show that a 2 × n checkerboard can be tiled with 2 × 1 dominoes in Fn+1 many ways. Solution: Denote the number of tilings of a 2 × n rectangle by Tn. We check that T1 = 1 and T2 = 2. We want to prove that they satisfy the recurrence relation Tn = Tn−1 + Tn−2: Consider the domino occupying the rightmost spot in the top row of the tiling. It is either a vertical domino, in which case the rest of the tiling can be interpreted as a tiling of a 2 × (n − 1) rectangle, or it is a horizontal domino, in which case there must be another horizontal domino under it, and the rest of the tiling can be interpreted as a tiling of a 2 × (n − 2) rectangle. Therefore Tn = Tn−1 + Tn−2: Since the number of tilings satisfies the same recurrence relation as the Fibonacci numbers, and T1 = F2 = 1 and T2 = F3 = 2, we may conclude that Tn = Fn+1. -
Remembrance of Alan Baker
Hardy-Ramanujan Journal 42 (2019), 4-11Hardy-Ramanujan 4-11 Remembrance of Alan Baker Gisbert W¨ustholz REMEMBRANCE OF ALAN BAKER It was at the conference on Diophantische Approximationen organized by Theodor Schneider by which took place at Oberwolfach Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter SchlickeweiGISBERT a graduate WUSTHOLZ¨ student of Schneider at the Albert Ludwigs University at Freiburg i. Br.. We both were invited to the conference to help Schneider organize the conference in practical matters. In particular we were determined to make the speakers write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for the future. Indeed if you open the webpage of the Mathematisches Forschungsinstitut Oberwolfach you find a link to the Oberwolfach Digital Archive where you can find all these Documents It was at the conference on Diophantische Approximationen organized by Theodor Schneider which took place at fromOberwofach 1944-2008. Juli 14 - Juli 20 1974 when I first met Alan Baker. At the time I was together with Hans Peter Schlickewei a graduateAt this student conference of Schneider I was at verythe Albert-Ludwig-University much impressed and at Freiburg touched i. Br.. by We seeing both were the invitedFields to Medallist the conference Baker whoseto help work Schneider I had to studied organize the in conference the past in year practical to matters.do the Inp-adic particular analogue we were determined of Baker's to make recent the versions speakers A write down an abstract of their talks in the Vortagsbuch which was considered to be an important historical document for sharpeningthe future. -
Cubic and Biquadratic Reciprocity
Rational (!) Cubic and Biquadratic Reciprocity Paul Pollack 2005 Ross Summer Mathematics Program It is ordinary rational arithmetic which attracts the ordinary man ... G.H. Hardy, An Introduction to the Theory of Numbers, Bulletin of the AMS 35, 1929 1 Quadratic Reciprocity Law (Gauss). If p and q are distinct odd primes, then q p 1 q 1 p = ( 1) −2 −2 . p − q We also have the supplementary laws: 1 (p 1)/2 − = ( 1) − , p − 2 (p2 1)/8 and = ( 1) − . p − These laws enable us to completely character- ize the primes p for which a given prime q is a square. Question: Can we characterize the primes p for which a given prime q is a cube? a fourth power? We will focus on cubes in this talk. 2 QR in Action: From the supplementary law we know that 2 is a square modulo an odd prime p if and only if p 1 (mod 8). ≡± Or take q = 11. We have 11 = p for p 1 p 11 ≡ (mod 4), and 11 = p for p 1 (mod 4). p − 11 6≡ So solve the system of congruences p 1 (mod 4), p (mod 11). ≡ ≡ OR p 1 (mod 4), p (mod 11). ≡− 6≡ Computing which nonzero elements mod p are squares and nonsquares, we find that 11 is a square modulo a prime p = 2, 11 if and only if 6 p 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44). ≡ q Observe that the p with p = 1 are exactly the primes in certain arithmetic progressions. -
Second Grade Unit 1: Extending Base Ten Understanding
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Second Grade Unit 1: Extending Base Ten Understanding These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Georgia Department of Education Georgia Standards of Excellence Framework GSE Extending Base Ten Understanding Unit 1 Unit 1: Extending Base Ten Understanding TABLE OF CONTENTS Overview……………………………………………………………………………..... 3 Standards for Mathematical Practice …………………………………………………. 5 Standards for Mathematical Content …………………………………………………. 6 Big Ideas ………………………………………………………………………………. 6 Essential Questions ……………………………………………………………………….. 7 Concepts and Skills to Maintain ………………………………………………………. 7 Strategies for Teaching and Learning ……………………………………………. 8 Selected Terms and Symbols ………………………………………………..………… 10 Task Types ………………………………………………………………...………….. 12 Task Descriptions …………………………………………………………………….. 13 Intervention Table………………….………………………………………………….. 15 Straws! Straws! Straws!.........................................................................… 16 Where Am I on the Number Line? …………………………………… 23 I Spy a Number ………………………………………………………. 31 Number Hop …………………………………………………………… 38 Place Value Play …………………………………………………….... 44 The Importance of Zero ………………………………………………. 51 Base Ten Pictures …………………………………………………….. 62 Building Base Ten Numbers …………………………………………. 69 What's My Number? …………………………………………………. 74 Capture the Caterpillar ……………………………………………….. 79 Formative Assessment Lesson ……………………………………….. 88 Fill the Bucket ……………………………………………………….. -
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]). -
Sums of Gauss, Jacobi, and Jacobsthai* One of the Primary
JOURNAL OF NUMBER THEORY 11, 349-398 (1979) Sums of Gauss, Jacobi, and JacobsthaI* BRUCE C. BERNDT Department of Mathematics, University of Illinois, Urbana, Illinois 61801 AND RONALD J. EVANS Department of Mathematics, University of California at San Diego, L.a Jolla, California 92093 Received February 2, 1979 DEDICATED TO PROFESSOR S. CHOWLA ON THE OCCASION OF HIS 70TH BIRTHDAY 1. INTRODUCTION One of the primary motivations for this work is the desire to evaluate, for certain natural numbers k, the Gauss sum D-l G, = c e277inkl~, T&=0 wherep is a prime withp = 1 (mod k). The evaluation of G, was first achieved by Gauss. The sums Gk for k = 3,4, 5, and 6 have also been studied. It is known that G, is a root of a certain irreducible cubic polynomial. Except for a sign ambiguity, the value of G4 is known. See Hasse’s text [24, pp. 478-4941 for a detailed treatment of G, and G, , and a brief account of G, . For an account of G, , see a paper of E. Lehmer [29]. In Section 3, we shall determine G, (up to two sign ambiguities). Using our formula for G, , the second author [18] has recently evaluated G,, (up to four sign ambiguities). We shall also evaluate G, , G,, , and Gz4 in terms of G, . For completeness, we include in Sections 3.1 and 3.2 short proofs of known results on G, and G 4 ; these results will be used frequently in the sequel. (We do not discuss G, , since elaborate computations are involved, and G, is not needed in the sequel.) While evaluations of G, are of interest in number theory, they also have * This paper was originally accepted for publication in the Rocky Mountain Journal of Mathematics. -
Linking Together Members of the Mathematical Carlos Rocha, University of Lisbon; Jean Taylor, Cour- Community from the US and Abroad
NEWSLETTER OF THE EUROPEAN MATHEMATICAL SOCIETY Features Epimorphism Theorem Prime Numbers Interview J.-P. Bourguignon Societies European Physical Society Research Centres ESI Vienna December 2013 Issue 90 ISSN 1027-488X S E European M M Mathematical E S Society Cover photo: Jean-François Dars Mathematics and Computer Science from EDP Sciences www.esaim-cocv.org www.mmnp-journal.org www.rairo-ro.org www.esaim-m2an.org www.esaim-ps.org www.rairo-ita.org Contents Editorial Team European Editor-in-Chief Ulf Persson Matematiska Vetenskaper Lucia Di Vizio Chalmers tekniska högskola Université de Versailles- S-412 96 Göteborg, Sweden St Quentin e-mail: [email protected] Mathematical Laboratoire de Mathématiques 45 avenue des États-Unis Zdzisław Pogoda 78035 Versailles cedex, France Institute of Mathematicsr e-mail: [email protected] Jagiellonian University Society ul. prof. Stanisława Copy Editor Łojasiewicza 30-348 Kraków, Poland Chris Nunn e-mail: [email protected] Newsletter No. 90, December 2013 119 St Michaels Road, Aldershot, GU12 4JW, UK Themistocles M. Rassias Editorial: Meetings of Presidents – S. Huggett ............................ 3 e-mail: [email protected] (Problem Corner) Department of Mathematics A New Cover for the Newsletter – The Editorial Board ................. 5 Editors National Technical University Jean-Pierre Bourguignon: New President of the ERC .................. 8 of Athens, Zografou Campus Mariolina Bartolini Bussi GR-15780 Athens, Greece Peter Scholze to Receive 2013 Sastra Ramanujan Prize – K. Alladi 9 (Math. Education) e-mail: [email protected] DESU – Universitá di Modena e European Level Organisations for Women Mathematicians – Reggio Emilia Volker R. Remmert C. Series ............................................................................... 11 Via Allegri, 9 (History of Mathematics) Forty Years of the Epimorphism Theorem – I-42121 Reggio Emilia, Italy IZWT, Wuppertal University [email protected] D-42119 Wuppertal, Germany P.