FACTORS AND PRIMES IN TWO SMARANDACHE SEQUENCES RALF W. STEPHAN Abstract. Using a personal computer and freely available software, the author factored
|
|
|
- Claud Houston
- 5 years ago
- Views:
Transcription
1 FACTORS AND PRIMES IN TWO SMARANDACHE SEQUENCES RALF W. STEPHAN Abstract. Using a personal computer and freely available software, the author factored some members of the Smarandache consecutive sequence and the reverse Smarandache sequence. Nearly complete factorizations are given up to Sm(80) and RSm(80). Both sequences were excessively searched for prime members, with only one prime found up to Sm(840) and RSm(750): RSm(82)= Introduction Both the Smarandache consecutive sequence, and the reverse Smarandache sequence are described in [S93]. Throughout this article, Sm(n) denotes the nth member of the consecutive sequence, and RSm(n) the nth member of the reverse sequence, e.g. Sm(11)= , and RSm(11)= The Fundamental Theorem of Arithmetic states that every n 2 N, n>1 can be written as a product p1p2p3 :::pk of a nite number of primes. This "factorization" is unique for n if the pk are ordered by size. A proof can be found in [R85]. Factorization of large numbers has rapidly advanced in the past decades, both through better algorithms and faster hardware. Although there is still no polynomialtime algorithm known for nding prime factors pk of composite numbers n = pk, Q several methods have been developed that allow factoring of numbers with 100 digits or more within reasonable time: the elliptic curve method (ECM) by Lenstra [L87], with enhancements by Montgomery [M87][M92] and others, has found factors with up to 49 digits, as of April Its running time depends on the size of the unknown p, and only slightly on the size of n. the quadratic sieve [S87] and the number eld sieve [LL93]. The running time of these methods depends on the size of n. Factors with digits are frequently found by NFSNet 1. For log p 50 and log n= log p 2, sieving methods are faster than ECM. Because ECM time depends on p, which is unknown from the start, it is dicult to predict when a factor will be found. Therefore, when fully factoring a large number, one tries to eliminate small factors rst, using conventional sieving and other methods, then one looks for factors with 20, 30, and 40 digits using ECM, and nally, if there is enough computing power, one of the sieving methods is applied. The primality ofthefactors and the remaining numbers is usually shown rst through a probabilistic test [K81] that has a small enough failure probability like Such a prime is called a probable prime. Proving primality can be done using number theory or the ECPP method by Atkin/Morain [AM93]. 1 URL: 1
2 2 RALF W. STEPHAN In the following, pn denotes a probable prime of n digits, Pn is a proven prime with n digits, and cn means a composite number with n digits. 2. Free software For computations with large numbers, it is not necessary to buy one of the well known Computer Algebra software packages like Maple or Mathematica. There are several multiprecision libraries freely available that can be used with the programming language C. The advantage of using one of these libraries is that they are usually by an order of magnitude faster than interpreted code when compared on the same machine [Z98]. For factoring, we used science0 2 and GMP-ECM 3.To write the program for nding prime members of Sm(n) and RSm(n), we used the GMP 4 multiprecision library. For proving primality of RSm(82), we used ECPP Factorization results We used science0 to eliminate small factors of Sm(n) and RSm(n) with 1 < n 80, and GMP-ECM to nd factors of up to about 40 digits. The system is a Pentium 200 MHz running Linux 6. The timings we measured for reducing the probability of a factor with specic size to 1=e are given in the following table: log p log n B1 curves time : minutes hours days Table 1. Time to nd p with probability 1 1=e on a Pentium 200 MHz using GMP-ECM under Linux All remaining composites were searched with ECM parameter B1=40000 and 200 curves were computed. Therefore, the probability of a remaining factor with less than 24 digits is less than 1=e. No primes were proven. The following tables list the results. 2 URL: 3 URL: 4 URL: 5 URL: 6 URL:
3 FACTORS AND PRIMES IN TWO SEQUENCES 3 n known factors of Sm(n) p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p52 continued...
4 4 RALF W. STEPHAN n known factors of Sm(n) p p p p p p p p p p p c c c c p p c c c c c p c c c c c c c c p c c c p115 Table 2. Factorizations of Sm(n), 1 <n80
5 FACTORS AND PRIMES IN TWO SEQUENCES 5 n known factors of RSm(n) p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p41 continued...
6 6 RALF W. STEPHAN n known factors of RSm(n) p p p p p p p p p p p p p p c p c p p p c p c p p p p p p c c c c c p c121 Table 3. Factorizations of RSm(n), 1 <n80 4. Searching for primes in Sm and RSm Using the GMP library, a fast C program was written to search for primes in Sm(n) and RSm(n). We used the Miller-Rabin [K81] test to check for compositeness.
7 FACTORS AND PRIMES IN TWO SEQUENCES 7 No primes were found in Sm(n), 1 <n<840, and only one probable prime in RSm(n), 1 <n<750, namely RSm(82)= ::: This number proved prime with ECPP. 5. Acknowledgements and contact information Thanks go to Paul Zimmermann for discussion and review of the paper. He also contributed one factor to the data. This work wouldn't have been possible without the open-source software provided by the respective authors: Richard Crandall (science0), Torbjorn Granlund (GMP), Paul Zimmermann (GMP-ECM), and Francois Morain (ECPP). The author can be reached at the address stephan@tmt.de and his homepage is at the URL References [AM93] A.O.L.Atkin and F.Morain: Elliptic curves and primality proving, Math. Comp. 60 (1993) [K81] Donald E. Knuth: The Art of Computer Programming, Vol. 2, Seminumerical Algorithms, 2nd ed, Addison-Wesley, 1981 [L87] H.W.Lenstra, Jr.: Factoring integers with elliptic curves, Annals of Mathematics (2) 126 (1987), [LL93] A.K.Lenstra and H.W.Lenstra, Jr. (eds.): The development of the number eld sieve, Notes in Mathematics 1554, Springer-Verlag, Berlin, 1993 [M87] Peter L. Montgomery: Speeding the Pollard and Elliptic Curve Methods of Factorization, Math. Comp. 48 (1987), [M92] Peter L. Montgomery: An FFT Extension of the Elliptic Curve Method of Factorization, Ph.D. dissertation, Mathematics, University of California at Los Angeles, 1992 [R85] Hans Riesel: Prime Numbers and Computer Methods for Factorization, Birkhuser Verlag, 1985 [S87] R.D.Silverman: The multiple polynomial quadratic sieve, Math. Comp. 48 (1987), [S93] F.Smarandache: Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993 [Z98] P.Zimmermann: Comparison of three public-domain multiprecision libraries: Bignum, Gmp and Pari
Factorization myths. D. J. Bernstein. Thanks to: University of Illinois at Chicago NSF DMS Alfred P. Sloan Foundation
Factorization myths D. J. Bernstein Thanks to: University of Illinois at Chicago NSF DMS 0140542 Alfred P. Sloan Foundation Sieving and 611 + for small : 1 2 2 3 4 2 2 3 5 6 2 3 5 7 7 8 2 2 2 9 3 3 10
Calculators will not be permitted on the exam. The numbers on the exam will be suitable for calculating by hand.
Midterm #: practice MATH Intro to Number Theory midterm: Thursday, Nov 7 Please print your name: Calculators will not be permitted on the exam. The numbers on the exam will be suitable for calculating
Number Theory and Security in the Digital Age
Number Theory and Security in the Digital Age Lola Thompson Ross Program July 21, 2010 Lola Thompson (Ross Program) Number Theory and Security in the Digital Age July 21, 2010 1 / 37 Introduction I have
Dedekind Sums with Arguments Near Euler s Number e
1 47 6 11 Journal of Integer Sequences, Vol. 1 (01), Article 1..8 Dedeind Sums with Arguments Near Euler s Number e Kurt Girstmair Institut für Mathemati Universität Innsbruc Technierstr. 1/7 A-600 Innsbruc
Q(173)Q(177)Q(188)Q(193)Q(203)
MATH 313: SOLUTIONS HW3 Problem 1 (a) 30941 We use the Miller-Rabin test to check if it prime. We know that the smallest number which is a strong pseudoprime both base 2 and base 3 is 1373653; hence, if
Number Theory. Konkreetne Matemaatika
ITT9131 Number Theory Konkreetne Matemaatika Chapter Four Divisibility Primes Prime examples Factorial Factors Relative primality `MOD': the Congruence Relation Independent Residues Additional Applications
On a Deconcatenation Problem
On a Deconcatenation Problem Henry Ibstedt Abstract: In a recent study of the Primality of the Smarandache Symmetric Sequences Sabin and Tatiana Tabirca [1] observed a very high frequency of the prime
EXPLAINING THE SHAPE OF RSK
EXPLAINING THE SHAPE OF RSK SIMON RUBINSTEIN-SALZEDO 1. Introduction There is an algorithm, due to Robinson, Schensted, and Knuth (henceforth RSK), that gives a bijection between permutations σ S n and
Pattern Avoidance in Unimodal and V-unimodal Permutations
Pattern Avoidance in Unimodal and V-unimodal Permutations Dido Salazar-Torres May 16, 2009 Abstract A characterization of unimodal, [321]-avoiding permutations and an enumeration shall be given.there is
Math 42, Discrete Mathematics
c Fall 2018 last updated 10/29/2018 at 18:22:13 For use by students in this class only; all rights reserved. Note: some prose & some tables are taken directly from Kenneth R. Rosen, and Its Applications,
Graphs of Tilings. Patrick Callahan, University of California Office of the President, Oakland, CA
Graphs of Tilings Patrick Callahan, University of California Office of the President, Oakland, CA Phyllis Chinn, Department of Mathematics Humboldt State University, Arcata, CA Silvia Heubach, Department
Cryptography, Number Theory, and RSA
Cryptography, Number Theory, and RSA Joan Boyar, IMADA, University of Southern Denmark November 2015 Outline Symmetric key cryptography Public key cryptography Introduction to number theory RSA Modular
THE ENUMERATION OF PERMUTATIONS SORTABLE BY POP STACKS IN PARALLEL
THE ENUMERATION OF PERMUTATIONS SORTABLE BY POP STACKS IN PARALLEL REBECCA SMITH Department of Mathematics SUNY Brockport Brockport, NY 14420 VINCENT VATTER Department of Mathematics Dartmouth College
Data security (Cryptography) exercise book
University of Debrecen Faculty of Informatics Data security (Cryptography) exercise book 1 Contents 1 RSA 4 1.1 RSA in general.................................. 4 1.2 RSA background.................................
The following code should by now seem familiar: do {
296 Chapter 7. Random Numbers if (n!= nold) { If n has changed, then compute useful quantities. en=n; oldg=gammln(en+1.0); nold=n; if (p!= pold) { If p has changed, then compute useful quantities. pc=1.0-p;
Assignment 2. Due: Monday Oct. 15, :59pm
Introduction To Discrete Math Due: Monday Oct. 15, 2012. 11:59pm Assignment 2 Instructor: Mohamed Omar Math 6a For all problems on assignments, you are allowed to use the textbook, class notes, and other
Rational Points On Elliptic Curves - Solutions. (i) Throughout, we ve been looking at elliptic curves in the general form. y 2 = x 3 + Ax + B
Rational Points On Elliptic Curves - Solutions (Send corrections to cbruni@uwaterloo.ca) (i) Throughout, we ve been looking at elliptic curves in the general form y 2 = x 3 + Ax + B However we did claim
Two congruences involving 4-cores
Two congruences involving 4-cores ABSTRACT. The goal of this paper is to prove two new congruences involving 4- cores using elementary techniques; namely, if a 4 (n) denotes the number of 4-cores of n,
Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite
Proof that Mersenne Prime Numbers are Infinite and that Even Perfect Numbers are Infinite Stephen Marshall 7 November 208 Abstract Mersenne prime is a prime number that is one less than a power of two.
Gauss and AGM. Burton Rosenberg. January 30, 2004
Gauss and AGM Burton Rosenberg January 3, 24 Introduction derivation of equation. what has it to do w/ the lemniscate agm properties of I elliptic integrals The Elliptic Integral of the First Kind Define
Introduction to Modular Arithmetic
1 Integers modulo n 1.1 Preliminaries Introduction to Modular Arithmetic Definition 1.1.1 (Equivalence relation). Let R be a relation on the set A. Recall that a relation R is a subset of the cartesian
THE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM
THE ASSOCIATION OF MATHEMATICS TEACHERS OF NEW JERSEY 2018 ANNUAL WINTER CONFERENCE FOSTERING GROWTH MINDSETS IN EVERY MATH CLASSROOM CREATING PRODUCTIVE LEARNING ENVIRONMENTS WEDNESDAY, FEBRUARY 7, 2018
Fast Sorting and Pattern-Avoiding Permutations
Fast Sorting and Pattern-Avoiding Permutations David Arthur Stanford University darthur@cs.stanford.edu Abstract We say a permutation π avoids a pattern σ if no length σ subsequence of π is ordered in
Circular Nim Games. S. Heubach 1 M. Dufour 2. May 7, 2010 Math Colloquium, Cal Poly San Luis Obispo
Circular Nim Games S. Heubach 1 M. Dufour 2 1 Dept. of Mathematics, California State University Los Angeles 2 Dept. of Mathematics, University of Quebeq, Montreal May 7, 2010 Math Colloquium, Cal Poly
Sheet 1: Introduction to prime numbers.
Option A Hand in at least one question from at least three sheets Sheet 1: Introduction to prime numbers. [provisional date for handing in: class 2.] 1. Use Sieve of Eratosthenes to find all prime numbers
MA/CSSE 473 Day 13. Student Questions. Permutation Generation. HW 6 due Monday, HW 7 next Thursday, Tuesday s exam. Permutation generation
MA/CSSE 473 Day 13 Permutation Generation MA/CSSE 473 Day 13 HW 6 due Monday, HW 7 next Thursday, Student Questions Tuesday s exam Permutation generation 1 Exam 1 If you want additional practice problems
EE 123 Discussion Section 6. Frank Ong March 14th, 2016
EE 123 Discussion Section 6 Frank Ong March 14th, 2016 Plan Sparse FFT Magnitude Filter Design with convex optimization Sparse FFT Given a length-n signal, FFT takes O(N log N) time to compute its DFT
Table of Contents. Table of Contents 1
Table of Contents 1) The Factor Game a) Investigation b) Rules c) Game Boards d) Game Table- Possible First Moves 2) Toying with Tiles a) Introduction b) Tiles 1-10 c) Tiles 11-16 d) Tiles 17-20 e) Tiles
Number Patterns - Grade 10 [CAPS] *
![Number Patterns - Grade 10 [CAPS] *](/thumbs/87/96010508.jpg "Number Patterns - Grade 10 [CAPS] *")
OpenStax-CNX module: m38376 1 Number Patterns - Grade 10 [CAPS] * Free High School Science Texts Project Based on Number Patterns by Rory Adams Free High School Science Texts Project Mark Horner Heather
MA/CSSE 473 Day 9. The algorithm (modified) N 1
MA/CSSE 473 Day 9 Primality Testing Encryption Intro The algorithm (modified) To test N for primality Pick positive integers a 1, a 2,, a k < N at random For each a i, check for a N 1 i 1 (mod N) Use the
Launchpad Maths. Arithmetic II
Launchpad Maths. Arithmetic II LAW OF DISTRIBUTION The Law of Distribution exploits the symmetries 1 of addition and multiplication to tell of how those operations behave when working together. Consider
Formulas for Primes. Eric Rowland Hofstra University. Eric Rowland Formulas for Primes / 27
Formulas for Primes Eric Rowland Hofstra University 2018 2 14 Eric Rowland Formulas for Primes 2018 2 14 1 / 27 The sequence of primes 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Bibliography. S. Gill Williamson
Bibliography S. Gill Williamson 1. S. G. Williamson, A Combinatorial Property of Finite Sequences with Applications to Tensor Algebra, J. Combinatorial Theory, 1 (1966), pp. 401-410. 2. S. G. Williamson,
Towards Real-time Hardware Gamma Correction for Dynamic Contrast Enhancement
Towards Real-time Gamma Correction for Dynamic Contrast Enhancement Jesse Scott, Ph.D. Candidate Integrated Design Services, College of Engineering, Pennsylvania State University University Park, PA jus2@engr.psu.edu
PRIMES IN SHIFTED SUMS OF LUCAS SEQUENCES. Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania
#A52 INTEGERS 17 (2017) PRIMES IN SHIFTED SUMS OF LUCAS SEQUENCES Lenny Jones Department of Mathematics, Shippensburg University, Shippensburg, Pennsylvania lkjone@ship.edu Lawrence Somer Department of
A Winning Strategy for the Game of Antonim
A Winning Strategy for the Game of Antonim arxiv:1506.01042v1 [math.co] 1 Jun 2015 Zachary Silbernick Robert Campbell June 4, 2015 Abstract The game of Antonim is a variant of the game Nim, with the additional
JMG. Review Module 1 Lessons 1-20 for Mid-Module. Prepare for Endof-Unit Assessment. Assessment. Module 1. End-of-Unit Assessment.
Lesson Plans Lesson Plan WEEK 161 December 5- December 9 Subject to change 2016-2017 Mrs. Whitman 1 st 2 nd Period 3 rd Period 4 th Period 5 th Period 6 th Period H S Mathematics Period Prep Geometry Math
Chapter 6.1. Cycles in Permutations
Chapter 6.1. Cycles in Permutations Prof. Tesler Math 184A Fall 2017 Prof. Tesler Ch. 6.1. Cycles in Permutations Math 184A / Fall 2017 1 / 27 Notations for permutations Consider a permutation in 1-line
Restricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers
Restricted Permutations Related to Fibonacci Numbers and k-generalized Fibonacci Numbers arxiv:math/0109219v1 [math.co] 27 Sep 2001 Eric S. Egge Department of Mathematics Gettysburg College 300 North Washington
phase switching in radio interferometry Eric Keto Smithsonian Astrophysical Observatory, 60 Garden Street,Cambridge, MA 02138
Shifted m-sequences as an alternative to Walsh functions for phase switching in radio interferometry Eric Keto Smithsonian Astrophysical Observatory, 60 Garden Street,Cambridge, MA 02138 Submillimeter
NUMBER THEORY AMIN WITNO
NUMBER THEORY AMIN WITNO.. w w w. w i t n o. c o m Number Theory Outlines and Problem Sets Amin Witno Preface These notes are mere outlines for the course Math 313 given at Philadelphia
Note Computations with a deck of cards
Theoretical Computer Science 259 (2001) 671 678 www.elsevier.com/locate/tcs Note Computations with a deck of cards Anton Stiglic Zero-Knowledge Systems Inc, 888 de Maisonneuve East, 6th Floor, Montreal,
Wilson s Theorem and Fermat s Theorem
Wilson s Theorem and Fermat s Theorem 7-27-2006 Wilson s theorem says that p is prime if and only if (p 1)! = 1 (mod p). Fermat s theorem says that if p is prime and p a, then a p 1 = 1 (mod p). Wilson
A4M33PAL, ZS , FEL ČVUT
Pseudorandom numbers John von Neumann: Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such
Alessandro Cincotti School of Information Science, Japan Advanced Institute of Science and Technology, Japan
#G03 INTEGERS 9 (2009),621-627 ON THE COMPLEXITY OF N-PLAYER HACKENBUSH Alessandro Cincotti School of Information Science, Japan Advanced Institute of Science and Technology, Japan cincotti@jaist.ac.jp
LUCAS-SIERPIŃSKI AND LUCAS-RIESEL NUMBERS
LUCAS-SIERPIŃSKI AND LUCAS-RIESEL NUMBERS DANIEL BACZKOWSKI, OLAOLU FASORANTI, AND CARRIE E. FINCH Abstract. In this paper, we show that there are infinitely many Sierpiński numbers in the sequence of
Distribution of Primes
Distribution of Primes Definition. For positive real numbers x, let π(x) be the number of prime numbers less than or equal to x. For example, π(1) = 0, π(10) = 4 and π(100) = 25. To use some ciphers, we
An Elementary Solution to the Ménage Problem
An Elementary Solution to the Ménage Problem Amanda F Passmore April 14, 2005 1 Introduction The ménage problem asks for the number of ways to seat n husbands and n wives at a circular table with alternating
arxiv: v1 [math.co] 30 Nov 2017
![arxiv: v1 [math.co] 30 Nov 2017](/thumbs/95/126343447.jpg "arxiv: v1 [math.co] 30 Nov 2017")
A NOTE ON 3-FREE PERMUTATIONS arxiv:1712.00105v1 [math.co] 30 Nov 2017 Bill Correll, Jr. MDA Information Systems LLC, Ann Arbor, MI, USA william.correll@mdaus.com Randy W. Ho Garmin International, Chandler,
Sequences. like 1, 2, 3, 4 while you are doing a dance or movement? Have you ever group things into
Math of the universe Paper 1 Sequences Kelly Tong 2017/07/17 Sequences Introduction Have you ever stamped your foot while listening to music? Have you ever counted like 1, 2, 3, 4 while you are doing a
A REMARK ON A PAPER OF LUCA AND WALSH 1. Zhao-Jun Li Department of Mathematics, Anhui Normal University, Wuhu, China. Min Tang 2.
#A40 INTEGERS 11 (2011) A REMARK ON A PAPER OF LUCA AND WALSH 1 Zhao-Jun Li Department of Mathematics, Anhui Normal University, Wuhu, China Min Tang 2 Department of Mathematics, Anhui Normal University,
Odd king tours on even chessboards
Odd king tours on even chessboards D. Joyner and M. Fourte, Department of Mathematics, U. S. Naval Academy, Annapolis, MD 21402 12-4-97 In this paper we show that there is no complete odd king tour on
Fermat s little theorem. RSA.
.. Computing large numbers modulo n (a) In modulo arithmetic, you can always reduce a large number to its remainder a a rem n (mod n). (b) Addition, subtraction, and multiplication preserve congruence:
Math236 Discrete Maths with Applications
Math236 Discrete Maths with Applications P. Ittmann UKZN, Pietermaritzburg Semester 1, 2012 Ittmann (UKZN PMB) Math236 2012 1 / 43 The Multiplication Principle Theorem Let S be a set of k-tuples (s 1,
Evacuation and a Geometric Construction for Fibonacci Tableaux
Evacuation and a Geometric Construction for Fibonacci Tableaux Kendra Killpatrick Pepperdine University 24255 Pacific Coast Highway Malibu, CA 90263-4321 Kendra.Killpatrick@pepperdine.edu August 25, 2004
An interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g.,
Binary exponentiation An interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g., What are the last two digits of the number 2 284? In the absence
PROOFS OF SOME BINOMIAL IDENTITIES USING THE METHOD OF LAST SQUARES
PROOFS OF SOME BINOMIAL IDENTITIES USING THE METHOD OF LAST SQUARES MARK SHATTUCK AND TAMÁS WALDHAUSER Abstract. We give combinatorial proofs for some identities involving binomial sums that have no closed
Synthetic Aperture Radar
Synthetic Aperture Radar J. Patrick Fitch Synthetic Aperture Radar C.S. Burrus, Consulting Editor With 93 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo J. Patrick Fitch Engineering
Zhanjiang , People s Republic of China
Math. Comp. 78(2009), no. 267, 1853 1866. COVERS OF THE INTEGERS WITH ODD MODULI AND THEIR APPLICATIONS TO THE FORMS x m 2 n AND x 2 F 3n /2 Ke-Jian Wu 1 and Zhi-Wei Sun 2, 1 Department of Mathematics,
Stacking Blocks and Counting Permutations
Stacking Blocks and Counting Permutations Lara K. Pudwell Valparaiso University Valparaiso, Indiana 46383 Lara.Pudwell@valpo.edu In this paper we will explore two seemingly unrelated counting questions,
A STUDY OF EULERIAN NUMBERS FOR PERMUTATIONS IN THE ALTERNATING GROUP
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A31 A STUDY OF EULERIAN NUMBERS FOR PERMUTATIONS IN THE ALTERNATING GROUP Shinji Tanimoto Department of Mathematics, Kochi Joshi University
Number-Theoretic Algorithms
Number-Theoretic Algorithms Hengfeng Wei hfwei@nju.edu.cn March 31 April 6, 2017 Hengfeng Wei (hfwei@nju.edu.cn) Number-Theoretic Algorithms March 31 April 6, 2017 1 / 36 Number-Theoretic Algorithms 1
Number Theory/Cryptography (part 1 of CSC 282)
Number Theory/Cryptography (part 1 of CSC 282) http://www.cs.rochester.edu/~stefanko/teaching/11cs282 1 Schedule The homework is due Sep 8 Graded homework will be available at noon Sep 9, noon. EXAM #1
30 HWASIN PARK, JOONGSOO PARK AND DAEYEOUL KIM Lemma 1.1. Let =2 k q +1, k 2 Z +. Then the set of rimitive roots modulo is the set of quadratic non-re
J. KSIAM Vol.4, No.1, 29-38, 2000 A CRITERION ON PRIMITIVE ROOTS MODULO Hwasin Park, Joongsoo Park and Daeyeoul Kim Abstract. In this aer, we consider a criterion on rimitive roots modulo where is the
RESTRICTED PERMUTATIONS AND POLYGONS. Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, Haifa, Israel
RESTRICTED PERMUTATIONS AND POLYGONS Ghassan Firro and Toufik Mansour Department of Mathematics, University of Haifa, 905 Haifa, Israel {gferro,toufik}@mathhaifaacil abstract Several authors have examined
Institute of Information and Mathematical Sciences. Prime Number Search Algorithms
Institute of Information and Mathematical Sciences Prime Number Search Algorithms (Mersenne Primes Search) 11191649 Jun Li June, 2012 Contents CHAPTER 1 INTRODUCTION... 1 1.1 BACKGROUND... 1 1.2 MERSENNE
Unique Sequences Containing No k-term Arithmetic Progressions
Unique Sequences Containing No k-term Arithmetic Progressions Tanbir Ahmed Department of Computer Science and Software Engineering Concordia University, Montréal, Canada ta ahmed@cs.concordia.ca Janusz
CS3334 Data Structures Lecture 4: Bubble Sort & Insertion Sort. Chee Wei Tan
CS3334 Data Structures Lecture 4: Bubble Sort & Insertion Sort Chee Wei Tan Sorting Since Time Immemorial Plimpton 322 Tablet: Sorted Pythagorean Triples https://www.maa.org/sites/default/files/pdf/news/monthly105-120.pdf
Ramanujan-type Congruences for Overpartitions Modulo 5. Nankai University, Tianjin , P. R. China
Ramanujan-type Congruences for Overpartitions Modulo 5 William Y.C. Chen a,b, Lisa H. Sun a,, Rong-Hua Wang a and Li Zhang a a Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071, P.
What we still don t know about addition and multiplication. Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA
What we still don t know about addition and multiplication Carl Pomerance, Dartmouth College Hanover, New Hampshire, USA You would think that all of the issues surrounding addition and multiplication were
ALGEBRA: Chapter I: QUESTION BANK
1 ALGEBRA: Chapter I: QUESTION BANK Elements of Number Theory Congruence One mark questions: 1 Define divisibility 2 If a b then prove that a kb k Z 3 If a b b c then PT a/c 4 If a b are two non zero integers
6. Find an inverse of a modulo m for each of these pairs of relatively prime integers using the method
Exercises Exercises 1. Show that 15 is an inverse of 7 modulo 26. 2. Show that 937 is an inverse of 13 modulo 2436. 3. By inspection (as discussed prior to Example 1), find an inverse of 4 modulo 9. 4.
Teacher s Notes. Problem of the Month: Courtney s Collection
Teacher s Notes Problem of the Month: Courtney s Collection Overview: In the Problem of the Month, Courtney s Collection, students use number theory, number operations, organized lists and counting methods
Pitch Detection Algorithms
OpenStax-CNX module: m11714 1 Pitch Detection Algorithms Gareth Middleton This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 1.0 Abstract Two algorithms to
Chapter 1. Probability
Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.
Implementation of an IFFT for an Optical OFDM Transmitter with 12.1 Gbit/s
Implementation of an IFFT for an Optical OFDM Transmitter with 12.1 Gbit/s Michael Bernhard, Joachim Speidel Universität Stuttgart, Institut für achrichtenübertragung, 7569 Stuttgart E-Mail: bernhard@inue.uni-stuttgart.de
Lossy Compression of Permutations
204 IEEE International Symposium on Information Theory Lossy Compression of Permutations Da Wang EECS Dept., MIT Cambridge, MA, USA Email: dawang@mit.edu Arya Mazumdar ECE Dept., Univ. of Minnesota Twin
Permutation Polynomials Modulo 2 w
Finite Fields and Their Applications 7, 287}292 (2001) doi.10.1006/!ta.2000.0282, available online at http://www.idealibrary.com on Permutation Polynomials Modulo 2 w Ronald L. Rivest Laboratory for Computer
Mathematical Language
NOTES Supplies Calculator Blank 3-by-5 index cards (at least 10 per student) Colored markers Mathematical Language Variable a quantity whose value changes or varies. A variable could also be defined as
ElGamal Public-Key Encryption and Signature
ElGamal Public-Key Encryption and Signature Çetin Kaya Koç koc@cs.ucsb.edu Çetin Kaya Koç http://koclab.org Winter 2017 1 / 10 ElGamal Cryptosystem and Signature Scheme Taher ElGamal, originally from Egypt,
Games of No Strategy and Low-Grade Combinatorics
Games of No Strategy and Low-Grade Combinatorics James Propp (jamespropp.org), UMass Lowell Mathematical Enchantments (mathenchant.org) presented at MOVES 2015 on August 3, 2015 Slides at http://jamespropp.org/moves15.pdf
Counting Permutations by Putting Balls into Boxes
Counting Permutations by Putting Balls into Boxes Ira M. Gessel Brandeis University C&O@40 Conference June 19, 2007 I will tell you shamelessly what my bottom line is: It is placing balls into boxes. Gian-Carlo
A theorem on the cores of partitions
A theorem on the cores of partitions Jørn B. Olsson Department of Mathematical Sciences, University of Copenhagen Universitetsparken 5,DK-2100 Copenhagen Ø, Denmark August 9, 2008 Abstract: If s and t
Generalized Game Trees
Generalized Game Trees Richard E. Korf Computer Science Department University of California, Los Angeles Los Angeles, Ca. 90024 Abstract We consider two generalizations of the standard two-player game
BAND SURGERY ON KNOTS AND LINKS, III
BAND SURGERY ON KNOTS AND LINKS, III TAIZO KANENOBU Abstract. We give two criteria of links concerning a band surgery: The first one is a condition on the determinants of links which are related by a band
Minimum Zero-Centre-Pandiagonal Composite Type II (a) Magic Squares over Multi Set of Integer Numbers as a Semiring
Minimum Zero-Centre-Pandiagonal Composite Type II (a) Magic Squares over Multi Set of Integer Numbers as a Semiring Babayo A.M. 1, Moharram Ali Khan 2 1. Department of Mathematics and Computer Science,
PAPER. Connecting the dots. Giovanna Roda Vienna, Austria
PAPER Connecting the dots Giovanna Roda Vienna, Austria giovanna.roda@gmail.com Abstract Symbolic Computation is an area of computer science that after 20 years of initial research had its acme in the
The Complexity of Sorting with Networks of Stacks and Queues
The Complexity of Sorting with Networks of Stacks and Queues Stefan Felsner Institut für Mathematik, Technische Universität Berlin. felsner@math.tu-berlin.de Martin Pergel Department of Applied Mathematics
DVA325 Formal Languages, Automata and Models of Computation (FABER)
DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November
Three of these grids share a property that the other three do not. Can you find such a property? + mod
PPMTC 22 Session 6: Mad Vet Puzzles Session 6: Mad Veterinarian Puzzles There is a collection of problems that have come to be known as "Mad Veterinarian Puzzles", for reasons which will soon become obvious.
1 Introduction The n-queens problem is a classical combinatorial problem in the AI search area. We are particularly interested in the n-queens problem
(appeared in SIGART Bulletin, Vol. 1, 3, pp. 7-11, Oct, 1990.) A Polynomial Time Algorithm for the N-Queens Problem 1 Rok Sosic and Jun Gu Department of Computer Science 2 University of Utah Salt Lake
8.3 Prove It! A Practice Understanding Task
15 8.3 Prove It! A Practice Understanding Task In this task you need to use all the things you know about quadrilaterals, distance, and slope to prove that the shapes are parallelograms, rectangles, rhombi,
Introduction. and Z r1 Z rn. This lecture aims to provide techniques. CRT during the decription process in RSA is explained.
THE CHINESE REMAINDER THEOREM INTRODUCED IN A GENERAL KONTEXT Introduction The rst Chinese problem in indeterminate analysis is encountered in a book written by the Chinese mathematician Sun Tzi. The problem
A CONJECTURE ON UNIT FRACTIONS INVOLVING PRIMES
Last update: Nov. 6, 2015. A CONJECTURE ON UNIT FRACTIONS INVOLVING PRIMES Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing 210093, People s Republic of China zwsun@nju.edu.cn http://math.nju.edu.cn/
Number Theory - Divisibility Number Theory - Congruences. Number Theory. June 23, Number Theory
- Divisibility - Congruences June 23, 2014 Primes - Divisibility - Congruences Definition A positive integer p is prime if p 2 and its only positive factors are itself and 1. Otherwise, if p 2, then p
Combinatorial Aspects of Flashcard Games
Combinatorial Aspects of Flashcard Games The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published Publisher Lewis, Joel
To be able to determine the quadratic character of an arbitrary number mod p (p an odd prime), we. The first (and most delicate) case concerns 2
Quadratic Reciprocity To be able to determine the quadratic character of an arbitrary number mod p (p an odd prime), we need to be able to evaluate q for any prime q. The first (and most delicate) case
An Aperiodic Tiling from a Dynamical System: An Exposition of An Example of Culik and Kari. S. Eigen J. Navarro V. Prasad
An Aperiodic Tiling from a Dynamical System: An Exposition of An Example of Culik and Kari S. Eigen J. Navarro V. Prasad These tiles can tile the plane But only Aperiodically Example A (Culik-Kari) Dynamical
Chapter 2 Channel Equalization
Chapter 2 Channel Equalization 2.1 Introduction In wireless communication systems signal experiences distortion due to fading [17]. As signal propagates, it follows multiple paths between transmitter and
Reflections on the N + k Queens Problem
Integre Technical Publishing Co., Inc. College Mathematics Journal 40:3 March 12, 2009 2:02 p.m. chatham.tex page 204 Reflections on the N + k Queens Problem R. Douglas Chatham R. Douglas Chatham (d.chatham@moreheadstate.edu)