Fermat Test with Gaussian Base and Gaussian Pseudoprimes 3
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Number Games Full Book.Pdf
Should you read this book? If you think it is interesting that on September 11, 2001, Flight 77 reportedly hit the 77 foot tall Pentagon, in Washington D.C. on the 77th Meridian West, after taking off at 8:20 AM and crashing at 9:37 AM, 77 minutes later, then this book is for you. Furthermore, if you can comprehend that there is a code of numbers behind the letters of the English language, as simple as A, B, C is 1, 2, 3, and using this code reveals that phrases and names such as ‘September Eleventh’, ‘World Trade Center’ and ‘Order From Chaos’ equate to 77, this book is definitely for you. And please know, these are all facts, just the same as it is a fact that Pentagon construction began September 11, 1941, just prior to Pearl Harbor. Table of Contents 1 – Introduction to Gematria, the Language of The Cabal 2 – 1968, Year of the Coronavirus & 9/11 Master Plan 3 – 222 Months Later, From 9/11 to the Coronavirus Pandemic 4 – Event 201, The Jesuit Order, Anthony Fauci & Pope Francis 5 – Crimson Contagion Pandemic Exercise & New York Times 6 – Clade X Pandemic Exercise & the Pandemic 666 Days Later 7 – Operation Dark Winter & Mr. Bright’s “Darkest Winter” Warning 8 – Donald Trump’s Vaccine Plan, Operation Warp Speed 9 – H.R. 6666, Contact Tracing, ID2020 & the Big Tech Takeover 10 – Rockefeller’s 2010 Scenarios for the Future of Technology 11 – Bill Gates’ First Birthday on Jonas Salk’s 42nd... & Elvis 12 – Tom Hanks & the Use of Celebrity to Sell the Pandemic 13 – Nadia the Tiger, Tiger King & Year of the Tiger, 2022 14 – Coronavirus Predictive -
International Refugee Assistance Project
General Docket United States Court of Appeals for the Fourth Circuit Court of Appeals Docket #: 17-1351 Docketed: 03/17/2017 Nature of Suit: 2440 Other Civil Rights Termed: 05/25/2017 Intl Refugee Assistance v. Donald J. Trump Appeal From: United States District Court for the District of Maryland at Greenbelt Fee Status: us Case Type Information: 1) Civil U.S. 2) United States 3) null Originating Court Information: District: 0416-8 : 8:17-cv-00361-TDC Court Reporter: Kathy Chiarizia, Court Reporter Coordinator (Grnblt) Court Reporter: Cindy Davis, Official Court Reporter Presiding Judge: Theodore D. Chuang, U. S. District Court Judge Date Filed: 02/07/2017 Date Date Order/Judgment Date NOA Date Rec'd Order/Judgment: EOD: Filed: COA: 03/16/2017 03/16/2017 03/17/2017 03/17/2017 Prior Cases: None Current Cases: None INTERNATIONAL REFUGEE Spencer Elijah Wittmann Amdur ASSISTANCE PROJECT, a project of the Direct: 212-549-2572 Urban Justice Center, Inc., on behalf of itself Email: [email protected] and its clients [COR NTC Retained] Plaintiff - Appellee AMERICAN CIVIL LIBERTIES UNION 18th Floor 125 Broad Street New York, NY 10004-2400 David Cole [On Brief] AMERICAN CIVIL LIBERTIES UNION 915 15th Street, NW Washington, DC 20005-0000 Justin Bryan Cox Direct: 678-404-9119 Email: [email protected] [COR NTC Retained] NATIONAL IMMIGRATION LAW CENTER 1989 College Avenue, NE Atlanta, GA 30317 Nicholas David Espiritu Email: [email protected] [COR NTC Retained] NATIONAL IMMIGRATION LAW CENTER 3435 Wilshire Boulevard Los Angeles, CA 90010 Lee P. Gelernt Direct: 212-549-2616 Email: [email protected] [COR NTC Retained] AMERICAN CIVIL LIBERTIES UNION 125 Broad Street New York, NY 10004-2400 Hugh Handeyside [On Brief] AMERICAN CIVIL LIBERTIES UNION 125 Broad Street New York, NY 10004-2400 Marielena Hincapie Direct: 213-639-3900 Email: [email protected] [COR NTC Retained] NATIONAL IMMIGRATION LAW CENTER Suite 1600 3435 Wilshire Boulevard Los Angeles, CA 90010 Omar C. -
Gaussian Prime Labeling of Super Subdivision of Star Graphs
of Math al em rn a u ti o c J s l A a Int. J. Math. And Appl., 8(4)(2020), 35{39 n n d o i i t t a s n A ISSN: 2347-1557 r e p t p n l I i c • Available Online: http://ijmaa.in/ a t 7 i o 5 n 5 • s 1 - 7 4 I 3 S 2 S : N International Journal of Mathematics And its Applications Gaussian Prime Labeling of Super Subdivision of Star Graphs T. J. Rajesh Kumar1,∗ and Antony Sanoj Jerome2 1 Department of Mathematics, T.K.M College of Engineering, Kollam, Kerala, India. 2 Research Scholar, University College, Thiruvananthapuram, Kerala, India. Abstract: Gaussian integers are the complex numbers of the form a + bi where a; b 2 Z and i2 = −1 and it is denoted by Z[i]. A Gaussian prime labeling on G is a bijection from the vertices of G to [ n], the set of the first n Gaussian integers in the spiral ordering such that if uv 2 E(G), then (u) and (v) are relatively prime. Using the order on the Gaussian integers, we discuss the Gaussian prime labeling of super subdivision of star graphs. MSC: 05C78. Keywords: Gaussian Integers, Gaussian Prime Labeling, Super Subdivision of Graphs. © JS Publication. 1. Introduction The graphs considered in this paper are finite and simple. The terms which are not defined here can be referred from Gallian [1] and West [2]. A labeling or valuation of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy, a label depending upon the vertex labels f(x) and f(y). -
Input for Carnival of Math: Number 115, October 2014
Input for Carnival of Math: Number 115, October 2014 I visited Singapore in 1996 and the people were very kind to me. So I though this might be a little payback for their kindness. Good Luck. David Brooks The “Mathematical Association of America” (http://maanumberaday.blogspot.com/2009/11/115.html ) notes that: 115 = 5 x 23. 115 = 23 x (2 + 3). 115 has a unique representation as a sum of three squares: 3 2 + 5 2 + 9 2 = 115. 115 is the smallest three-digit integer, abc , such that ( abc )/( a*b*c) is prime : 115/5 = 23. STS-115 was a space shuttle mission to the International Space Station flown by the space shuttle Atlantis on Sept. 9, 2006. The “Online Encyclopedia of Integer Sequences” (http://www.oeis.org) notes that 115 is a tridecagonal (or 13-gonal) number. Also, 115 is the number of rooted trees with 8 vertices (or nodes). If you do a search for 115 on the OEIS website you will find out that there are 7,041 integer sequences that contain the number 115. The website “Positive Integers” (http://www.positiveintegers.org/115) notes that 115 is a palindromic and repdigit number when written in base 22 (5522). The website “Number Gossip” (http://www.numbergossip.com) notes that: 115 is the smallest three-digit integer, abc, such that (abc)/(a*b*c) is prime. It also notes that 115 is a composite, deficient, lucky, odd odious and square-free number. The website “Numbers Aplenty” (http://www.numbersaplenty.com/115) notes that: It has 4 divisors, whose sum is σ = 144. -
Mathematical Circus & 'Martin Gardner
MARTIN GARDNE MATHEMATICAL ;MATH EMATICAL ASSOCIATION J OF AMERICA MATHEMATICAL CIRCUS & 'MARTIN GARDNER THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, DC 1992 MATHEMATICAL More Puzzles, Games, Paradoxes, and Other Mathematical Entertainments from Scientific American with a Preface by Donald Knuth, A Postscript, from the Author, and a new Bibliography by Mr. Gardner, Thoughts from Readers, and 105 Drawings and Published in the United States of America by The Mathematical Association of America Copyright O 1968,1969,1970,1971,1979,1981,1992by Martin Gardner. All riglhts reserved under International and Pan-American Copyright Conventions. An MAA Spectrum book This book was updated and revised from the 1981 edition published by Vantage Books, New York. Most of this book originally appeared in slightly different form in Scientific American. Library of Congress Catalog Card Number 92-060996 ISBN 0-88385-506-2 Manufactured in the United States of America For Donald E. Knuth, extraordinary mathematician, computer scientist, writer, musician, humorist, recreational math buff, and much more SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications ANDREW STERRETT, JR.,Chairman Spectrum Editorial Board ROGER HORN, Chairman SABRA ANDERSON BART BRADEN UNDERWOOD DUDLEY HUGH M. EDGAR JEANNE LADUKE LESTER H. LANGE MARY PARKER MPP.a (@ SPECTRUM Also by Martin Gardner from The Mathematical Association of America 1529 Eighteenth Street, N.W. Washington, D. C. 20036 (202) 387- 5200 Riddles of the Sphinx and Other Mathematical Puzzle Tales Mathematical Carnival Mathematical Magic Show Contents Preface xi .. Introduction Xlll 1. Optical Illusions 3 Answers on page 14 2. Matches 16 Answers on page 27 3. -
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000)
A NEW LARGEST SMITH NUMBER Patrick Costello Department of Mathematics and Statistics, Eastern Kentucky University, Richmond, KY 40475 (Submitted September 2000) 1. INTRODUCTION In 1982, Albert Wilansky, a mathematics professor at Lehigh University wrote a short article in the Two-Year College Mathematics Journal [6]. In that article he identified a new subset of the composite numbers. He defined a Smith number to be a composite number where the sum of the digits in its prime factorization is equal to the digit sum of the number. The set was named in honor of Wi!anskyJs brother-in-law, Dr. Harold Smith, whose telephone number 493-7775 when written as a single number 4,937,775 possessed this interesting characteristic. Adding the digits in the number and the digits of its prime factors 3, 5, 5 and 65,837 resulted in identical sums of42. Wilansky provided two other examples of numbers with this characteristic: 9,985 and 6,036. Since that time, many things have been discovered about Smith numbers including the fact that there are infinitely many Smith numbers [4]. The largest Smith numbers were produced by Samuel Yates. Using a large repunit and large palindromic prime, Yates was able to produce Smith numbers having ten million digits and thirteen million digits. Using the same large repunit and a new large palindromic prime, the author is able to find a Smith number with over thirty-two million digits. 2. NOTATIONS AND BASIC FACTS For any positive integer w, we let S(ri) denote the sum of the digits of n. -
Number Theory Course Notes for MA 341, Spring 2018
Number Theory Course notes for MA 341, Spring 2018 Jared Weinstein May 2, 2018 Contents 1 Basic properties of the integers 3 1.1 Definitions: Z and Q .......................3 1.2 The well-ordering principle . .5 1.3 The division algorithm . .5 1.4 Running times . .6 1.5 The Euclidean algorithm . .8 1.6 The extended Euclidean algorithm . 10 1.7 Exercises due February 2. 11 2 The unique factorization theorem 12 2.1 Factorization into primes . 12 2.2 The proof that prime factorization is unique . 13 2.3 Valuations . 13 2.4 The rational root theorem . 15 2.5 Pythagorean triples . 16 2.6 Exercises due February 9 . 17 3 Congruences 17 3.1 Definition and basic properties . 17 3.2 Solving Linear Congruences . 18 3.3 The Chinese Remainder Theorem . 19 3.4 Modular Exponentiation . 20 3.5 Exercises due February 16 . 21 1 4 Units modulo m: Fermat's theorem and Euler's theorem 22 4.1 Units . 22 4.2 Powers modulo m ......................... 23 4.3 Fermat's theorem . 24 4.4 The φ function . 25 4.5 Euler's theorem . 26 4.6 Exercises due February 23 . 27 5 Orders and primitive elements 27 5.1 Basic properties of the function ordm .............. 27 5.2 Primitive roots . 28 5.3 The discrete logarithm . 30 5.4 Existence of primitive roots for a prime modulus . 30 5.5 Exercises due March 2 . 32 6 Some cryptographic applications 33 6.1 The basic problem of cryptography . 33 6.2 Ciphers, keys, and one-time pads . -
The Entropy Conundrum: a Solution Proposal
OPEN ACCESS www.sciforum.net/conference/ecea-1 Conference Proceedings Paper – Entropy The Entropy Conundrum: A Solution Proposal Rodolfo A. Fiorini 1,* 1 Politecnico di Milano, Department of Electronics, Information and Bioengineering, Milano, Italy; E- Mail: [email protected] * E-Mail: [email protected]; Tel.: +039-02-2399-3350; Fax: +039-02-2399-3360. Received: 11 September 2014 / Accepted: 11 October 2014 / Published: 3 November 2014 Abstract: In 2004, physicist Mark Newman, along with biologist Michael Lachmann and computer scientist Cristopher Moore, showed that if electromagnetic radiation is used as a transmission medium, the most information-efficient format for a given 1-D signal is indistinguishable from blackbody radiation. Since many natural processes maximize the Gibbs- Boltzmann entropy, they should give rise to spectra indistinguishable from optimally efficient transmission. In 2008, computer scientist C.S. Calude and physicist K. Svozil proved that "Quantum Randomness" is not Turing computable. In 2013, academic scientist R.A. Fiorini confirmed Newman, Lachmann and Moore's result, creating analogous example for 2-D signal (image), as an application of CICT in pattern recognition and image analysis. Paradoxically if you don’t know the code used for the message you can’t tell the difference between an information-rich message and a random jumble of letters. This is an entropy conundrum to solve. Even the most sophisticated instrumentation system is completely unable to reliably discriminate so called "random noise" from any combinatorially optimized encoded message, which CICT called "deterministic noise". Entropy fundamental concept crosses so many scientific and research areas, but, unfortunately, even across so many different disciplines, scientists have not yet worked out a definitive solution to the fundamental problem of the logical relationship between human experience and knowledge extraction. -
CONTINUANTS and SOME DECOMPOSITIONS INTO SQUARES 3 So, in Virtue of Smith’S Construction, Rather Than Computing the Whole Sequence We Need to Obtain
CONTINUANTS AND SOME DECOMPOSITIONS INTO SQUARES CHARLES DELORME AND GUILLERMO PINEDA-VILLAVICENCIO Abstract. In 1855 H. J. S. Smith [8] proved Fermat’s two-square theorem using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number p as a sum of two squares, given a solution of z2+1 0 (mod p), and vice versa. In this paper, we extend the use of≡ continuants to proper representations by sums of two squares in rings of polynomials over fields of characteristic different from 2. New deterministic algorithms for finding the corresponding proper representations are presented. Our approach will provide a new constructive proof of the four- square theorem and new proofs for other representations of integers by quaternary quadratic forms. 1. Introduction Fermat’s two-square theorem has always captivated the mathemat- ical community. Equally captivating are the known proofs of such a theorem; see, for instance, [4, 5, 8, 21, 39, 53]. Among these proofs we were enchanted by Smith’s elementary approach [8], which is well within the reach of undergraduates. We remark Smith’s proof is very similar to Hermite’s [21], Serret’s [39], and Brillhart’s [5]. Two main ingredients of Smith’s proof are the notion of continuant (Definition 2 for arbitrary rings) and the famous Euclidean algorithm. Let us recall here, for convenience, a definition taken from [25, pp. 148] Definition 1. Euclidean rings are rings R with no zero divisors which arXiv:1112.4535v3 [math.NT] 6 Aug 2014 are endowed with a Euclidean function N from R to the nonnegative integers such that for all τ , τ R with τ = 0, there exists q,r R 1 2 ∈ 1 6 ∈ such that τ2 = qτ1 + r and N(r) < N(τ1). -
Williams Meet Matrices Base B on A-Sets
Malaya Journal of Matematik, Vol. 8, No. 4, 1691-1693, 2020 https://doi.org/10.26637/MJM0804/0062 Williams meet matrices base b on A-sets N. Elumalai1 and R. Kalpana2* Abstract In this study Williams meet matrices base b = 3 on A-sets are considered to study the results based on structure theorem. The determinant and inverse of the matrix are also found. In addition the matrix is expressed in terms of the other two matrices. Keywords Meet matrices, Williams meet matrices base b, A-sets. AMS Subject Classification 15A09, 03G10. 1,2P.G. and Research Department of Mathematics, A.V.C. College (Autonomous)(Affiliated to Bharathidhasan University,Trichy), Mannampandal, Mayiladuthurai-609305, Tamil Nadu, India. *Corresponding author: [email protected] Article History: Received 24 July 2020; Accepted 17 September 2020 c 2020 MJM. Contents 2. Preliminaries In this section, we recall some definitions which are very 1 Introduction......................................1691 useful to this work. 2 Preliminaries.....................................1691 1. If S1 and S2 are nonempty subsets of P then S1 u S2 = 3 Theorems on Williams number base b = 2. 1691 [x ^ y=x 2 Sl;y 2 S2;x 6= yg where u is a binary opera- 4 Results on Williams meet matrices base b on A-sets tion. 1692 2. S = fx1;x2;::::::::::::xng ⊂ P with xi < x j ) i < j 5 Conclusion.......................................1692 and let A = fal;a2;::::::::::::an−1g be a multichain References.......................................1692 with a1 ≤ a2 ≤ :::::::::: ≤ an−1 The set S is said to be an A -set if fxkg u fxk+1;::::::;xng = fakg for all k = 1;2;::::::;n − 1. -
THE GAUSSIAN INTEGERS Since the Work of Gauss, Number Theorists
THE GAUSSIAN INTEGERS KEITH CONRAD Since the work of Gauss, number theorists have been interested in analogues of Z where concepts from arithmetic can also be developed. The example we will look at in this handout is the Gaussian integers: Z[i] = fa + bi : a; b 2 Zg: Excluding the last two sections of the handout, the topics we will study are extensions of common properties of the integers. Here is what we will cover in each section: (1) the norm on Z[i] (2) divisibility in Z[i] (3) the division theorem in Z[i] (4) the Euclidean algorithm Z[i] (5) Bezout's theorem in Z[i] (6) unique factorization in Z[i] (7) modular arithmetic in Z[i] (8) applications of Z[i] to the arithmetic of Z (9) primes in Z[i] 1. The Norm In Z, size is measured by the absolute value. In Z[i], we use the norm. Definition 1.1. For α = a + bi 2 Z[i], its norm is the product N(α) = αα = (a + bi)(a − bi) = a2 + b2: For example, N(2 + 7i) = 22 + 72 = 53. For m 2 Z, N(m) = m2. In particular, N(1) = 1. Thinking about a + bi as a complex number, its norm is the square of its usual absolute value: p ja + bij = a2 + b2; N(a + bi) = a2 + b2 = ja + bij2: The reason we prefer to deal with norms on Z[i] instead of absolute values on Z[i] is that norms are integers (rather than square roots), and the divisibility properties of norms in Z will provide important information about divisibility properties in Z[i]. -
Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases
INTERSECTIONS OF DELETED DIGITS CANTOR SETS WITH GAUSSIAN INTEGER BASES Athesissubmittedinpartialfulfillment of the requirements for the degree of Master of Science By VINCENT T. SHAW B.S., Wright State University, 2017 2020 Wright State University WRIGHT STATE UNIVERSITY SCHOOL OF GRADUATE STUDIES May 1, 2020 I HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER MY SUPER- VISION BY Vincent T. Shaw ENTITLED Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases BE ACCEPTED IN PARTIAL FULFILLMENT OF THE RE- QUIREMENTS FOR THE DEGREE OF Master of Science. ____________________ Steen Pedersen, Ph.D. Thesis Director ____________________ Ayse Sahin, Ph.D. Department Chair Committee on Final Examination ____________________ Steen Pedersen, Ph.D. ____________________ Qingbo Huang, Ph.D. ____________________ Anthony Evans, Ph.D. ____________________ Barry Milligan, Ph.D. Interim Dean, School of Graduate Studies Abstract Shaw, Vincent T. M.S., Department of Mathematics and Statistics, Wright State University, 2020. Intersections of Deleted Digits Cantor Sets with Gaussian Integer Bases. In this paper, the intersections of deleted digits Cantor sets and their fractal dimensions were analyzed. Previously, it had been shown that for any dimension between 0 and the dimension of the given deleted digits Cantor set of the real number line, a translate of the set could be constructed such that the intersection of the set with the translate would have this dimension. Here, we consider deleted digits Cantor sets of the complex plane with Gaussian integer bases and show that the result still holds. iii Contents 1 Introduction 1 1.1 WhystudyintersectionsofCantorsets? . 1 1.2 Prior work on this problem . 1 2 Negative Base Representations 5 2.1 IntegerRepresentations .............................