DOCSLIB.ORG

Covering Systems: Number Theory in the Spirit of Paul Erd˝Os

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Covering Systems: Number Theory in the Spirit of Paul Erd˝Os Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Covering systems: Part 1. About Paul Erdos˝ number theory in the spirit of Paul Erdos˝ Part 2. Erdos˝ and covering systems in research Part 3. An Mark Kozek Erdos˝ problem on covering systems Whittier College Part 4: Sierpinski´ and Riesel December 7, 2010 revisited About Paul Erdos.˝ Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Part 1. About Paul Erdos˝ Part 2. Erdos˝ and covering systems in research Part 3. An Erdos˝ problem on covering systems Part 4: Sierpinski´ and Hungarian mathematician, 1913-1996. Riesel revisited Interested in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, probability theory... More on Paul Erdos.˝ Covering systems: Wrote/co-wrote over 1400 research papers. number theory in the spirit of Paul Erdos˝ Erdos˝ number. Mark Kozek Erdos˝ problems: issued “bounties” for problems that he Part 1. About thought were interesting or for which he wanted to Paul Erdos˝ know the solution. Part 2. Erdos˝ and covering systems in Erdos˝ resources: research 1 The man who loved only numbers, book by Paul Part 3. An Erdos˝ problem Hoffman, 1998, (based on his Atlantic Monthly article on covering systems from 1989). Part 4: 2 Sierpinski´ and N is a Number: A portrait of Paul Erdos˝ , film by George Riesel revisited Csicsery, 1993 (also based on the Atlantic Monthly article). 3 And what is your Erdos˝ number?", Caspar Goffman, the American Mathematical Monthly, 1969. Erdos˝ and covering systems in research Covering systems: number theory in the spirit of Paul Erdos˝ Question: (de Polignac, 1849) Mark Kozek Is it the case that every sufficiently large odd integer > 1 can Part 1. About be written as the sum of a prime number and a power of 2? Paul Erdos˝ Part 2. Erdos˝ and covering Some small counter examples include: 127, 905. systems in research (Romanoff) A positive proportion of the integers may be Part 3. An Erdos˝ problem expressed this way. on covering systems (van der Corput) The exceptions form a set of positive Part 4: density. Sierpinski´ and Riesel revisited (Erdos)˝ Constructed an arithmetic progression of odd integers not representable in this way. Background: (from Number Theory 101) Covering systems: Def.: Congruent number theory in the spirit of The integers a and b are congruent modulo the natural Paul Erdos˝ Mark Kozek number n > 1 if there exists an integer, z such that Part 1. About Paul Erdos˝ a − b = zn. Part 2. Erdos˝ and covering If so, we write systems in research a ≡ b (mod n). Part 3. An Erdos˝ problem on covering For example, systems Part 4: 77 ≡ 7 (mod 10) Sierpinski´ and Riesel 500 ≡ 1700 (mod 1200) revisited 3 ≡ 58 (mod 5) 143 ≡ 0 (mod 11). Background: (from Number Theory 101) Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek Def.: Covering system Part 1. About Paul Erdos˝ A covering system or covering, for short, is a finite system of Part 2. Erdos˝ congruences and covering systems in research n ≡ ai (mod mi ), 1 < i ≤ t, Part 3. An Erdos˝ problem on covering systems such that every integer satisfies at least one of the Part 4: congruences. Sierpinski´ and Riesel revisited Example of a covering. Covering systems: number theory in the spirit of Paul Erdos˝ Mark Kozek For example, the congruences Part 1. About Paul Erdos˝ n ≡ 0 (mod 2) Part 2. Erdos˝ ≡ ( ) and covering n 1 mod 3 systems in research n ≡ 3 (mod 4) Part 3. An n ≡ 5 (mod 6) Erdos˝ problem on covering systems n ≡ 9 (mod 12) Part 4: Sierpinski´ and Riesel form a covering of the integers. revisited More background. Covering systems: Def.: Chinese Remainder Theorem number theory in the spirit of Paul Erdos˝ A system of congruences has a unique solution if the moduli Mark Kozek are pairwise relatively prime. Part 1. About Paul Erdos˝ For example: Can we solve for x in the following system? Part 2. Erdos˝ and covering x ≡ 3 mod 9 systems in research x ≡ 5 mod 10 Part 3. An x ≡ 2 mod 11 Erdos˝ problem on covering systems Part 4: Sierpinski´ and Yes, because {9,10,11} are pair-wise, relatively prime. We Riesel revisited get: x ≡ 255 mod 990 (Note: 990 = 9 × 10 × 11) Back to de Polignac’s question. Covering systems: Question: (de Polignac, 1849) number theory in the spirit of > Paul Erdos˝ Is it the case that every sufficiently large odd integer 1 can Mark Kozek be written as the sum of a prime number and a power of 2? Part 1. About Paul Erdos˝ de Polignac asked about odd numbers of the form Part 2. Erdos˝ n and covering x = p + 2 . systems in research Erdos˝ instead thought about writing primes in the form Part 3. An Erdos˝ problem n on covering p = x − 2 , systems Part 4: and changed the question to: Sierpinski´ and Riesel revisited Question0: (Erdos)˝ Is it possible to find an integer x such that x − 2n is not prime for all (non-negative) integers n? Erdos’˝ approach to de Polignac’s conjecture Covering systems: This led to the “sub-question”: For which n and for which x number theory n in the spirit of would x − 2 be divisible by 3? Paul Erdos˝ Mark Kozek ⇔ x − 2n = 3z ⇔ x − 2n ≡ 0 (mod 3) ⇔ x ≡ 2n (mod 3) Part 1. About Paul Erdos˝ Let us take powers of 2 (mod 3) Part 2. Erdos˝ and covering 0 systems in 2 ≡ 1 (mod 3) research 21 ≡ 2 (mod 3) Part 3. An 2 Erdos˝ problem 2 ≡ 4 ≡ 1 (mod 3) on covering 3 systems 2 ≡ 8 ≡ 2 (mod 3) Part 4: Sierpinski´ and We see that: Riesel revisited 2n 6≡ 0 (mod 3). For even powers of 2, 2n ≡ 1 (mod 3). For odd powers of 2, 2n ≡ 2 (mod 3). Erdos’˝ approach to de Polignac’s conjecture Covering systems: number theory in the spirit of Paul Erdos˝ This led to the “sub-question”: For which n and for which x Mark Kozek would x − 2n be divisible by 3? Part 1. About Paul Erdos˝ We have that if, Part 2. Erdos˝ and covering systems in n research n ≡ 0 (mod 2) (even) and x ≡ 1 (mod 3) then x − 2 is Part 3. An divisible by 3. Erdos˝ problem on covering systems OR Part 4: n Sierpinski´ and n ≡ 1 (mod 2) (odd) and x ≡ 2 (mod 3) then x − 2 is Riesel revisited divisible by 3. Erdos’˝ approach to de Polignac’s conjecture Covering systems: number theory in the spirit of Erdos’˝ strategy was to continue along these lines and try to Paul Erdos˝ find conditions on n and on x that would ensure that x − 2n Mark Kozek would be divisible by primes from a given set (that would Part 1. About include 3). Paul Erdos˝ Part 2. Erdos˝ He found the following relations: and covering systems in n research n ≡ 0 (mod 2) and x ≡ 1 (mod 3) ⇒ 3|x − 2 Part 3. An n Erdos˝ problem n ≡ 0 (mod 3) and x ≡ 1 (mod 7) ⇒ 7|x − 2 on covering n systems n ≡ 1 (mod 4) and x ≡ 2 (mod 5) ⇒ 5|x − 2 Part 4: n Sierpinski´ and n ≡ 3 (mod 8) and x ≡ 8 (mod 17) ⇒ 17|x − 2 Riesel n revisited n ≡ 7 (mod 12) and x ≡ 11 (mod 13) ⇒ 13|x − 2 n ≡ 23 (mod 4) and x ≡ 121 (mod 241) ⇒ 241|x − 2n Erdos’˝ approach to de Polignac’s conjecture Covering systems: number theory n in the spirit of n ≡ 0 (mod 2) and x ≡ 1 (mod 3) ⇒ 3|x − 2 Paul Erdos˝ n Mark Kozek n ≡ 0 (mod 3) and x ≡ 1 (mod 7) ⇒ 7|x − 2 n Part 1. About n ≡ 1 (mod 4) and x ≡ 2 (mod 5) ⇒ 5|x − 2 Paul Erdos˝ n ≡ 3 (mod 8) and x ≡ 8 (mod 17) ⇒ 17|x − 2n Part 2. Erdos˝ and covering n systems in n ≡ 7 (mod 12) and x ≡ 11 (mod 13) ⇒ 13|x − 2 research n ≡ 23 (mod 24) and x ≡ 121 Part 3. An n Erdos˝ problem (mod 241) ⇒ 241|x − 2 on covering systems We observe that the set of congruences describing n form a Part 4: Sierpinski´ and covering system. Riesel revisited We observe that the set of congruences describing x can be combined solved using Chinese Remainder Theorem. Erdos’˝ arithmetic progression of counterexamples Covering systems: number theory in the spirit of Paul Erdos˝ By the Chinese Remainder Theorem, we get Mark Kozek Part 1. About x ≡ 7629217 (mod 11184810), Paul Erdos˝ Part 2. Erdos˝ and covering where systems in research 11184810 = 2 · 3 · 5 · 7 · 13 · 17 · 241. Part 3. An Erdos˝ problem This gives us the Erdos’˝ arithmetic progression of on covering systems counterexamples to de Polignac’s conjecture. Part 4: Sierpinski´ and k = 7629217 ± 11184810z, for z ∈ . Riesel Z revisited Erdos’˝ arithmetic progression of counterexamples Covering systems: number theory in the spirit of There exists an arithmetic progression of odd integers x, Paul Erdos˝ that simultaneously satisfy the system of congruences Mark Kozek Part 1. About x ≡ 1 (mod 3) Paul Erdos˝ x ≡ 1 (mod 7) Part 2. Erdos˝ and covering x ≡ 2 (mod 5) systems in research x ≡ 8 (mod 17) Part 3. An x ≡ 11 (mod 13) Erdos˝ problem on covering x ≡ 121 (mod 241) systems x ≡ 1 (mod 2), Part 4: Sierpinski´ and n Riesel such that x − 2 is composite (not prime) for all non-negative revisited integers n because x − 2n will be divisible by at least one of the primes from the set {3, 5, 7, 13, 17, 241}.
Load more

Recommended publications
  • Read the Table of Square Roots 1-1000 Txt for Amazon
    NCERT Solutions for Class 10 Maths Chapter 12. Numbers In Words 1 50 Number Words Chart Two Worksheets. Do Not Sell My Info (for CA Residents). When autocomplete results are available use up and down arrows to review and enter to select. Touch device users, explore by touch or with swipe gestures. Generated by PureJoy. Creation: 16:14 - May 21, 2020. Last Modification: 00:11 - Aug 29, 2020. If a 2 is present in ternary, turn it into 1T. For example, Your email address will not be published. Required fields are marked *. in the A440 pitch standard, a bit more than an octave higher in pitch than the highest note on a standard piano. Raising a physical quantity to a power results in a new physical quantity having a dimension with the exponents multiplied by the power. cubic meter = cubic feet x 0.028317017 cubic feet = cubic meter x. The MKS system based on the meter, kilogram second was augmented to allow force and energy from electrical quantities to be measured in one rationalized system of units. +91-22-25116741 The system was proposed by Giorgi in 1904. It was adopted by the IEC in 1935 to take [email protected] effect on January 1, 1940. The electrical to mechanical conversion was chosen to be based on the permeability of free space to be. 10111– palindromic prime in bases 3 (111212111 3 ) and 27 (DND 27 ). There are both color and black and white versions of the charts in printable pdf form. "10,000 promises" is a song by the Backstreet Boys.
    [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]
  • Sierpiński and Carmichael Numbers
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 1, January 2015, Pages 355–376 S 0002-9947(2014)06083-2 Article electronically published on September 23, 2014 SIERPINSKI´ AND CARMICHAEL NUMBERS WILLIAM BANKS, CARRIE FINCH, FLORIAN LUCA, CARL POMERANCE, AND PANTELIMON STANIC˘ A˘ Abstract. We establish several related results on Carmichael, Sierpi´nski and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael for every n ∈ N.Next, using a recent result of Matom¨aki and Wright, we show that there are x1/5 Carmichael numbers up to x that are also Sierpi´nski and Riesel. Finally, we show that if 2nk + 1 is Lehmer, then n 150 ω(k)2 log k,whereω(k)isthe number of distinct primes dividing k. 1. Introduction In 1960, Sierpi´nski [25] showed that there are infinitely many odd natural num- bers k with the property that 2nk + 1 is composite for every natural number n; such an integer k is called a Sierpi´nski number in honor of his work. Two years later, J. Selfridge (unpublished) showed that 78557 is a Sierpi´nski number, and this is still the smallest known example.1 Every currently known Sierpi´nski number k possesses at least one covering set P, which is a finite set of prime numbers with the property that 2nk + 1 is divisible by some prime in P for every n ∈ N.
    [Show full text]
  • Primegrid: Searching for a New World Record Prime Number
    PrimeGrid: Searching for a New World Record Prime Number rimeGrid [1] is a volunteer computing project which has mid-19th century the only known method of primality proving two main aims; firstly to find large prime numbers, and sec- was to exhaustively trial divide the candidate integer by all primes Pondly to educate members of the project and the wider pub- up to its square root. With some small improvements due to Euler, lic about the mathematics of primes. This means engaging people this method was used by Fortuné Llandry in 1867 to prove the from all walks of life in computational mathematics is essential to primality of 3203431780337 (13 digits long). Only 9 years later a the success of the project. breakthrough was to come when Édouard Lucas developed a new In the first regard we have been very successful – as of No- method based on Group Theory, and proved 2127 – 1 (39 digits) to vember 2013, over 70% of the primes on the Top 5000 list [2] of be prime. Modified slightly by Lehmer in the 1930s, Lucas Se- largest known primes were discovered by PrimeGrid. The project quences are still in use today! also holds various records including the discoveries of the largest The next important breakthrough in primality testing was the known Cullen and Woodall Primes (with a little over 2 million development of electronic computers in the latter half of the 20th and 1 million decimal digits, respectively), the largest known Twin century. In 1951 the largest known prime (proved with the aid Primes and Sophie Germain Prime Pairs, and the longest sequence of a mechanical calculator), was (2148 + 1)/17 at 49 digits long, of primes in arithmetic progression (26 of them, with a difference but this was swiftly beaten by several successive discoveries by of over 23 million between each).
    [Show full text]
  • Some Computational Experiments in Number Theory
    Some computational experiments in number theory Wieb Bosma Mathematisch Instituut Radboud University Nijmegen Nijmegen, the Netherlands [email protected] Summary. The Magma code and some computational results of experiments in number theory are given. The experiments concern covering systems with applica- tions to explicit primality tests, the inverse of Euler’s totient function, and class number relations in Galois extensions of Q. Some evidence for various conjectures and open problems is given. 1 Introduction In the course of 10 years of working with and for Magma [6], I have conducted a large number of computational experiments in number theory. Many of them were meant, at least initially, as tests for new algorithms or implementations. In this paper I have collected results from, and code for, a few of those exper- iments. Three main themes can be recognized in the material below: covering sys- tems, the Euler φ function, and class number relations. In section 3 it is shown how covering systems can be used, with cubic reciprocity, to produce a simple criterion for the primality of n = h 3k + 1 in terms of a cubic recurrence modulo n; the starting value depends only· on the residue class of k modulo some covering modulus M. These primality tests generalize the Lucas–Lehmer type tests for numbers of the form h 2k + 1. They lead to a question about values of h for which h 3k + 1 is composite· for every k, and a generalization of a problem of Sierpi´nski.· We found a 12- digit number h with this property — a candidate analogue for the number 78577, which is most likely the smallest h with the property that h 2k +1 is composite for every k.
    [Show full text]
  • Colloquium Mathematicum
    C O L L O Q U I U M .... M A T H E M A T I C U M nnoindentVOL periodCOLLOQUIUM 86 .... 2 0 .... NO periodn 1h f i l l MATHEMATICUM INFINITE FAMILIES OF NONCOTOTIENTS nnoindentBY VOL . 86 n h f i l l 2 0 n h f i l l NO . 1 A period .. F L A M M E N K A M P AND F period .. L U C A .. open parenthesis BIELEFELD closing parenthesisn centerline fINFINITECOLLOQUIUMMATHEMATICUM FAMILIES OF NONCOTOTIENTS g Abstract periodVOL For .any 86 positive integer n let phi open parenthesis 2 0 n closing parenthesis be the NOEuler . 1 function n centerline fBY g of n period A positive INFINITE FAMILIES OF NONCOTOTIENTS integer n is called a noncototient if the equation x minus phi open parenthesis x closing parenthesis = n has n centerline fA. nquad FLAMMENKAMPANDFBY . nquad LUCA nquad ( BIELEFELD ) g no solution x period .. InA.FLAMMENKAMP AND F . L U C A ( BIELEFELD ) this note comma weAbstract give a sufficient . For any condition positive integer on a positiven let φ integer(n) be kthe such Euler that function the geometrical of n: A positive n hspace ∗fn f i l l g Abstract . For any positive integer $ n $ let $ nphi ( n ) $ progression openinteger parenthesisn is called 2 to a the noncototient power of if m the k closing equation parenthesisx − φ(x) = subn has m greater no solution equalx: 1 consistsIn this entirely be the Euler function of $ n . $ A positive of noncototients periodnote , .. we We give then a sufficient use computations condition on to a positive integer k such that the geometrical detect seven suchprogression positive integers(2mk) k≥ period1 consists entirely of noncototients .
    [Show full text]
  • Elliot R. Emadian Department of Dance, University of Illinois, Urbana, Illinois
    #A72 INTEGERS 18 (2018) RUTH-AARON PAIRS CONTAINING RIESEL OR SIERPINSKI´ NUMBERS Elliot R. Emadian Department of Dance, University of Illinois, Urbana, Illinois Carrie E. Finch-Smith Department of Mathematics, Washington and Lee University, Lexington, Virginia [email protected] Margaret G. Kallus Department of Mathematics, Washington and Lee University, Lexington, Virginia Received: 3/15/17, Revised: 7/18/18, Accepted: 8/17/18, Published: 8/31/18 Abstract In this paper, we construct a Ruth-Aaron pair that contains a Riesel number. We also construct a Ruth-Aaron pair that contains a Sierpinski´ number. Finally, we construct a Ruth-Aaron pair with the smaller number in the pair a Riesel number and the larger number in the pair a Sierpinski´ number. 1. Introduction On April 8, 1974, Hank Aaron hit his 715th career home run in Atlanta, Georgia, breaking Babe Ruth’s longstanding record of 714 career home runs. Soon after, mathematicians at the University of Georgia noticed that 714 and 715 were an interesting pair of numbers. We have the factorizations 2 3 7 17 = 714 and 5 11 13 = 715. · · · · · We see that the sum of the factors is the same for these numbers: 2 + 3 + 7 + 17 = 29 and 5 + 11 + 13 = 29. From this, Ruth-Aaron pairs are defined. For a positive integer n with prime factorization n = pe1 pe2 pei , we define S(n) = e p + e p + + e p . If 1 · 2 · · · i 1 1 2 2 · · · i i S(n) = S(n + 1), then n and n + 1 form a Ruth-Aaron pair.
    [Show full text]
  • CANT 2018 Abstracts
    CANT 2018 Abstracts Sixteenth Annual Workshop on Combinatorial and Additive Number Theory CUNY Graduate Center May 22{25, 2018 Sukumar Das Adhikari, Harish-Chandra Research Institute, and Ramakrishna Mission Vivekananda Educational and Research Institute, India Title: Some early Ramsey-type theorems in combinatorial number theory and some generalizations Abstract: We start with some classical results in Ramsey Theory and report some recent results on some questions related to monochromatic solutions of linear Dio- phantine equations. We shall also mention few challenging open questions in this area of Ramsey Theory. Shabnam Akhtari, University of Oregon Title: Representation of integers by binary forms Abstract: Let F (x; y) be an irreducible binary form of degree at least 3 and with integer coefficients. By a well-known result of Thue, the equation F (x; y) = m has only finitely many solutions in integers x and y. I will discuss some quantitative results on the number of solutions of such equations. I will also talk about my recent joint work with Manjul Bhargava, where we show that many equations of the shape F (x; y) = m have no solutions. Paul Baginski, Fairfield University Title: Nonunique factorization in the ring of integer-valued polynomials Abstract: The ring of integer-valued polynomials Int(Z) is the set of polynomials with rational coefficients which produce integer values for integer inputs. Specifi- cally, Int(Z) = ff(x) 2 Q[x] j 8n 2 Z f(n) 2 Zg: Int(Z) constitutes an interesting example in algebra from many perspectives; for example, it is a natural example of a non-Noetherian ring.
    [Show full text]
  • Subject Index
    Subject Index Many of these terms are defined in the glossary, others are defined in the Prime Curios! themselves. The boldfaced entries should indicate the key entries. γ 97 Arecibo Message 55 φ 79, 184, see golden ratio arithmetic progression 34, 81, π 8, 12, 90, 102, 106, 129, 136, 104, 112, 137, 158, 205, 210, 154, 164, 172, 173, 177, 181, 214, 219, 223, 226, 227, 236 187, 218, 230, 232, 235 Armstrong number 215 5TP39 209 Ars Magna 20 ASCII 66, 158, 212, 230 absolute prime 65, 146, 251 atomic number 44, 51, 64, 65 abundant number 103, 156 Australopithecus afarensis 46 aibohphobia 19 autism 85 aliquot sequence 13, 98 autobiographical prime 192 almost-all-even-digits prime 251 averaging sets 186 almost-equipandigital prime 251 alphabet code 50, 52, 61, 65, 73, Babbage 18, 146 81, 83 Babbage (portrait) 147 alphaprime code 83, 92, 110 balanced prime 12, 48, 113, 251 alternate-digit prime 251 Balog 104, 159 Amdahl Six 38 Balog cube 104 American Mathematical Society baseball 38, 97, 101, 116, 127, 70, 102, 196, 270 129 Antikythera mechanism 44 beast number 109, 129, 202, 204 apocalyptic number 72 beastly prime 142, 155, 229, 251 Apollonius 101 bemirp 113, 191, 210, 251 Archimedean solid 19 Bernoulli number 84, 94, 102 Archimedes 25, 33, 101, 167 Bernoulli triangle 214 { Page 287 { Bertrand prime Subject Index Bertrand prime 211 composite-digit prime 59, 136, Bertrand's postulate 111, 211, 252 252 computer mouse 187 Bible 23, 45, 49, 50, 59, 72, 83, congruence 252 85, 109, 158, 194, 216, 235, congruent prime 29, 196, 203, 236 213, 222, 227,
    [Show full text]
  • Recurrence Sequences Mathematical Surveys and Monographs Volume 104
    http://dx.doi.org/10.1090/surv/104 Recurrence Sequences Mathematical Surveys and Monographs Volume 104 Recurrence Sequences Graham Everest Alf van der Poorten Igor Shparlinski Thomas Ward American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Michael P. Loss Peter S. Landweber, Chair Tudor Stefan Ratiu J. T. Stafford 2000 Mathematics Subject Classification. Primary 11B37, 11B39, 11G05, 11T23, 33B10, 11J71, 11K45, 11B85, 37B15, 94A60. For additional information and updates on this book, visit www.ams.org/bookpages/surv-104 Library of Congress Cataloging-in-Publication Data Recurrence sequences / Graham Everest... [et al.]. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 104) Includes bibliographical references and index. ISBN 0-8218-3387-1 (alk. paper) 1. Recurrent sequences (Mathematics) I. Everest, Graham, 1957- II. Series. QA246.5.R43 2003 512/.72—dc21 2003050346 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government.
    [Show full text]
  • Dictionary of Mathematics
    Dictionary of Mathematics English – Spanish | Spanish – English Diccionario de Matemáticas Inglés – Castellano | Castellano – Inglés Kenneth Allen Hornak Lexicographer © 2008 Editorial Castilla La Vieja Copyright 2012 by Kenneth Allen Hornak Editorial Castilla La Vieja, c/o P.O. Box 1356, Lansdowne, Penna. 19050 United States of America PH: (908) 399-6273 e-mail: [email protected] All dictionaries may be seen at: http://www.EditorialCastilla.com Sello: Fachada de la Universidad de Salamanca (ESPAÑA) ISBN: 978-0-9860058-0-0 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any informational storage or retrieval system without permission in writing from the author Kenneth Allen Hornak. Reservados todos los derechos. Quedan rigurosamente prohibidos la reproducción de este libro, el tratamiento informático, la transmisión de alguna forma o por cualquier medio, ya sea electrónico, mecánico, por fotocopia, por registro u otros medios, sin el permiso previo y por escrito del autor Kenneth Allen Hornak. ACKNOWLEDGEMENTS Among those who have favoured the author with their selfless assistance throughout the extended period of compilation of this dictionary are Andrew Hornak, Norma Hornak, Edward Hornak, Daniel Pritchard and T.S. Gallione. Without their assistance the completion of this work would have been greatly delayed. AGRADECIMIENTOS Entre los que han favorecido al autor con su desinteresada colaboración a lo largo del dilatado período de acopio del material para el presente diccionario figuran Andrew Hornak, Norma Hornak, Edward Hornak, Daniel Pritchard y T.S. Gallione. Sin su ayuda la terminación de esta obra se hubiera demorado grandemente.
    [Show full text]
  • Polygonal, Sierpinski, and Riesel Numbers
    1 2 Journal of Integer Sequences, Vol. 18 (2015), 3 Article 15.8.1 47 6 23 11 Polygonal, Sierpi´nski, and Riesel numbers Daniel Baczkowski and Justin Eitner Department of Mathematics The University of Findlay Findlay, OH 45840 USA [email protected] [email protected] Carrie E. Finch1 and Braedon Suminski2 Department of Mathematics Washington and Lee University Lexington, VA 24450 USA [email protected] [email protected] Mark Kozek Department of Mathematics Whittier College Whittier, CA 90608 USA [email protected] Abstract In this paper, we show that there are infinitely many Sierpi´nski numbers in the se- quence of triangular numbers, hexagonal numbers, and pentagonal numbers. We also show that there are infinitely many Riesel numbers in the same sequences. Further- more, we show that there are infinitely many n-gonal numbers that are simultaneously Sierpi´nski and Riesel. 1Partially supported by a Lenfest Grant. 2Supported by the R. E. Lee Summer Scholar program. 1 1 Introduction Polygonal numbers are those that can be expressed geometrically by an arrangement of equally spaced points. For example, a positive integer n is a triangular number if n dots can be arranged in the form of an equilateral triangle. Similarly, n is a square number if n dots can be arranged in the form of a square. The diagram below represents the first four hexagonal numbers, which are 1, 6, 15, and 28. In 1960, Sierpi´nski [11] showed that there are infinitely many odd positive integers k with the property that k ·2n +1 is composite for all positive integers n.
    [Show full text]