A Quantum Primality Test with Order Finding
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Simultaneous Elements of Prescribed Multiplicative Orders 2
Simultaneous Elements Of Prescribed Multiplicative Orders N. A. Carella Abstract: Let u 6= ±1, and v 6= ±1 be a pair of fixed relatively prime squarefree integers, and let d ≥ 1, and e ≥ 1 be a pair of fixed integers. It is shown that there are infinitely many primes p ≥ 2 such that u and v have simultaneous prescribed multiplicative orders ordp u = (p − 1)/d and ordp v = (p − 1)/e respectively, unconditionally. In particular, a squarefree odd integer u > 2 and v = 2 are simultaneous primitive roots and quadratic residues (or quadratic nonresidues) modulo p for infinitely many primes p, unconditionally. Contents 1 Introduction 1 2 Divisors Free Characteristic Function 4 3 Finite Summation Kernels 5 4 Gaussian Sums, And Weil Sums 7 5 Incomplete And Complete Exponential Sums 7 6 Explicit Exponential Sums 11 7 Equivalent Exponential Sums 12 8 Double Exponential Sums 13 9 The Main Term 14 10 The Error Terms 15 arXiv:2103.04822v1 [math.GM] 5 Mar 2021 11 Simultaneous Prescribed Multiplicative Orders 16 12 Probabilistic Results For Simultaneous Orders 18 1 Introduction The earliest study of admissible integers k-tuples u1, u2, . , uk ∈ Z of simultaneous prim- itive roots modulo a prime p seems to be the conditional result in [21]. Much more general March 9, 2021 AMS MSC2020 :Primary 11A07; Secondary 11N13 Keywords: Distribution of primes; Primes in arithmetic progressions; Simultaneous primitive roots; Simul- taneous prescribed orders; Schinzel-Wojcik problem. 1 Simultaneous Elements Of Prescribed Multiplicative Orders 2 results for admissible rationals k-tuples u1, u2, . , uk ∈ Q of simultaneous elements of independent (or pseudo independent) multiplicative orders modulo a prime p ≥ 2 are considered in [26], and [14]. -
Computing Multiplicative Order and Primitive Root in Finite Cyclic Group
Computing Multiplicative Order and Primitive Root in Finite Cyclic Group Shri Prakash Dwivedi ∗ Abstract Multiplicative order of an element a of group G is the least positive integer n such that an = e, where e is the identity element of G. If the order of an element is equal to |G|, it is called generator or primitive root. This paper describes the algorithms for computing multiplicative order and Z∗ primitive root in p, we also present a logarithmic improvement over classical algorithms. 1 Introduction Algorithms for computing multiplicative order or simply order and primitive root or generator are important in the area of random number generation and discrete logarithm problem among oth- Z∗ ers. In this paper we consider the problem of computing the order of an element over p, which is Z∗ multiplicative group modulo p . As stated above order of an element a ∈ p is the least positive integer n such that an = 1. In number theoretic language, we say that the order of a modulo m is n, if n is the smallest positive integer such that an ≡ 1( mod m). We also consider the related Z∗ Z∗ problem of computing the primitive root in p. If order of an element a ∈ n usually denoted as Z∗ ordn(a) is equal to | n| i.e. order of multiplicative group modulo n, then a is called called primi- Z∗ tive root or primitive element or generator [3] of n. It is called generator because every element Z∗ in n is some power of a. Efficient deterministic algorithms for both of the above problems are not known yet. -
Grade 7 Mathematics Strand 6 Pattern and Algebra
GR 7 MATHEMATICS S6 1 TITLE PAGE GRADE 7 MATHEMATICS STRAND 6 PATTERN AND ALGEBRA SUB-STRAND 1: NUMBER PATTERNS SUB-STRAND 2: DIRECTED NUMBERS SUB-STRAND 3: INDICES SUB-STARND 4: ALGEBRA GR 7 MATHEMATICS S6 2 ACKNOWLEDGEMENT Acknowledgements We acknowledge the contributions of all Secondary and Upper Primary Teachers who in one way or another helped to develop this Course. Special thanks to the Staff of the mathematics Department of FODE who played active role in coordinating writing workshops, outsourcing lesson writing and editing processes, involving selected teachers of Madang, Central Province and NCD. We also acknowledge the professional guidance provided by the Curriculum Development and Assessment Division throughout the processes of writing and, the services given by the members of the Mathematics Review and Academic Committees. The development of this book was co-funded by GoPNG and World Bank. MR. DEMAS TONGOGO Principal- FODE . Written by: Luzviminda B. Fernandez SCO-Mathematics Department Flexible Open and Distance Education Papua New Guinea Published in 2016 @ Copyright 2016, Department of Education Papua New Guinea All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means electronic, mechanical, photocopying, recording or any other form of reproduction by any process is allowed without the prior permission of the publisher. ISBN: 978 - 9980 - 87 - 250 - 0 National Library Services of Papua New Guinea Printed by the Flexible, Open and Distance Education GR 7 MATHEMATICS S6 3 CONTENTS CONTENTS Page Secretary‟s Message…………………………………….…………………………………......... 4 Strand Introduction…………………………………….…………………………………………. 5 Study Guide………………………………………………….……………………………………. 6 SUB-STRAND 1: NUMBER PATTERNS ……………...….……….……………..……….. -
Tutor Talk Binary Numbers
What are binary numbers and why do we use them? BINARY NUMBERS Converting decimal numbers to binary numbers The number system we commonly use is decimal numbers, also known as Base 10. Ones, tens, hundreds, and thousands. For example, 4351 represents 4 thousands, 3 hundreds, 5 tens, and 1 ones. Thousands Hundreds Tens ones 4 3 5 1 Thousands Hundreds Tens ones 4 3 5 1 However, a computer does not understand decimal numbers. It only understands “on and off,” “yes and no.” Thousands Hundreds Tens ones 4 3 5 1 In order to convey “yes and no” to a computer, we use the numbers one (“yes” or “on”) and zero (“no” or “off”). To break it down further, the number 4351 represents 1 times 1, 5 times 10, DECIMAL NUMBERS (BASE 10) 3 times 100, and 4 times 1000. Each step to the left is another multiplication of 10. This is why it is called Base 10, or decimal numbers. The prefix dec- 4351 means ten. 4x1000 3x100 5x10 1x1 One is 10 to the zero power. Anything raised to the zero power is one. DECIMAL NUMBERS (BASE 10) Ten is 10 to the first power (or 10). One hundred is 10 to the second power (or 10 times 10). One thousand is 10 to the third 4351 power (or 10 times 10 times 10). 4x1000 3x100 5x10 1x1 103=1000 102=100 101=10 100=1 Binary numbers, or Base 2, use the number 2 instead of the number 10. 103 102 101 100 The prefix bi- means two. -
Exponential and Character Sums with Mersenne Numbers
Exponential and character sums with Mersenne numbers William D. Banks Dept. of Mathematics, University of Missouri Columbia, MO 65211, USA [email protected] John B. Friedlander Dept. of Mathematics, University of Toronto Toronto, Ontario M5S 3G3, Canada [email protected] Moubariz Z. Garaev Instituto de Matem´aticas Universidad Nacional Aut´onoma de M´exico C.P. 58089, Morelia, Michoac´an, M´exico [email protected] Igor E. Shparlinski Dept. of Computing, Macquarie University Sydney, NSW 2109, Australia [email protected] Dedicated to the memory of Alf van der Poorten 1 Abstract We give new bounds on sums of the form Λ(n) exp(2πiagn/m) and Λ(n)χ(gn + a), 6 6 nXN nXN where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums exp(2πiaMp/m) and χ(Mp) 6 6 pXN pXN p with Mersenne numbers Mp = 2 − 1, where p is prime. AMS Subject Classification Numbers: 11L07, 11L20. 1 Introduction Let m be an arbitrary natural number, and let a and g be integers that are coprime to m. Our aim in the present note is to give bounds on exponential sums and multiplicative character sums of the form n n Sm(a; N)= Λ(n)em(ag ) and Tm(χ, a; N)= Λ(n)χ(g + a), 6 6 nXN nXN where em is the additive character modulo m defined by em(x) = exp(2πix/m) (x ∈ R), and χ is a nontrivial multiplicative character modulo m. -
Enciclopedia Matematica a Claselor De Numere Întregi
THE MATH ENCYCLOPEDIA OF SMARANDACHE TYPE NOTIONS vol. I. NUMBER THEORY Marius Coman INTRODUCTION About the works of Florentin Smarandache have been written a lot of books (he himself wrote dozens of books and articles regarding math, physics, literature, philosophy). Being a globally recognized personality in both mathematics (there are countless functions and concepts that bear his name), it is natural that the volume of writings about his research is huge. What we try to do with this encyclopedia is to gather together as much as we can both from Smarandache’s mathematical work and the works of many mathematicians around the world inspired by the Smarandache notions. Because this is too vast to be covered in one book, we divide encyclopedia in more volumes. In this first volume of encyclopedia we try to synthesize his work in the field of number theory, one of the great Smarandache’s passions, a surfer on the ocean of numbers, to paraphrase the title of the book Surfing on the ocean of numbers – a few Smarandache notions and similar topics, by Henry Ibstedt. We quote from the introduction to the Smarandache’work “On new functions in number theory”, Moldova State University, Kishinev, 1999: “The performances in current mathematics, as the future discoveries, have, of course, their beginning in the oldest and the closest of philosophy branch of nathematics, the number theory. Mathematicians of all times have been, they still are, and they will be drawn to the beaty and variety of specific problems of this branch of mathematics. Queen of mathematics, which is the queen of sciences, as Gauss said, the number theory is shining with its light and attractions, fascinating and facilitating for us the knowledge of the laws that govern the macrocosm and the microcosm”. -
Overpseudoprimes, Mersenne Numbers and Wieferich Primes 2
OVERPSEUDOPRIMES, MERSENNE NUMBERS AND WIEFERICH PRIMES VLADIMIR SHEVELEV Abstract. We introduce a new class of pseudoprimes - so-called “overpseu- doprimes” which is a special subclass of super-Poulet pseudoprimes. De- noting via h(n) the multiplicative order of 2 modulo n, we show that odd number n is overpseudoprime if and only if the value of h(n) is invariant of all divisors d > 1 of n. In particular, we prove that all composite Mersenne numbers 2p − 1, where p is prime, and squares of Wieferich primes are overpseudoprimes. 1. Introduction n Sometimes the numbers Mn =2 − 1, n =1, 2,..., are called Mersenne numbers, although this name is usually reserved for numbers of the form p (1) Mp =2 − 1 where p is prime. In our paper we use the latter name. In this form numbers Mp at the first time were studied by Marin Mersenne (1588-1648) at least in 1644 (see in [1, p.9] and a large bibliography there). We start with the following simple observation. Let n be odd and h(n) denote the multiplicative order of 2 modulo n. arXiv:0806.3412v9 [math.NT] 15 Mar 2012 Theorem 1. Odd d> 1 is a divisor of Mp if and only if h(d)= p. Proof. If d > 1 is a divisor of 2p − 1, then h(d) divides prime p. But h(d) > 1. Thus, h(d)= p. The converse statement is evident. Remark 1. This observation for prime divisors of Mp belongs to Max Alek- seyev ( see his comment to sequence A122094 in [5]). -
Introduction to Abstract Algebra “Rings First”
Introduction to Abstract Algebra \Rings First" Bruno Benedetti University of Miami January 2020 Abstract The main purpose of these notes is to understand what Z; Q; R; C are, as well as their polynomial rings. Contents 0 Preliminaries 4 0.1 Injective and Surjective Functions..........................4 0.2 Natural numbers, induction, and Euclid's theorem.................6 0.3 The Euclidean Algorithm and Diophantine Equations............... 12 0.4 From Z and Q to R: The need for geometry..................... 18 0.5 Modular Arithmetics and Divisibility Criteria.................... 23 0.6 *Fermat's little theorem and decimal representation................ 28 0.7 Exercises........................................ 31 1 C-Rings, Fields and Domains 33 1.1 Invertible elements and Fields............................. 34 1.2 Zerodivisors and Domains............................... 36 1.3 Nilpotent elements and reduced C-rings....................... 39 1.4 *Gaussian Integers................................... 39 1.5 Exercises........................................ 41 2 Polynomials 43 2.1 Degree of a polynomial................................. 44 2.2 Euclidean division................................... 46 2.3 Complex conjugation.................................. 50 2.4 Symmetric Polynomials................................ 52 2.5 Exercises........................................ 56 3 Subrings, Homomorphisms, Ideals 57 3.1 Subrings......................................... 57 3.2 Homomorphisms.................................... 58 3.3 Ideals......................................... -
1 Integers and Divisibility
Jay Daigle Occidental College Math 322: Number Theory 1 Integers and Divisibility In this course we primarily want to study the factorization properties of integers. So we should probably start by reminding ourselves how integers and factorization work. Much of this material was covered in Math 210, but we shall review it so we can use it during the rest of the course, as well as perhaps putting it on a somewhat firmer foundation. 1.1 The integers and the rationals For further reading on the material in this subsection, consult Rosen 1.1{1.3; PMF 1.1{ 1.2, 2.1{2.2. Definition 1.1. The integers are elements of the set Z = f:::; −2; −1; 0; 1; 2;::: g. The natural numbers are elements of the set N = f1; 2;::: g of positive integers. The rational numbers are elements of the set Q = fp=q : p; q 2 Zg. Remark 1.2. 1. Some sources include 0 as a natural number; in this course we will not, and none of the four suggested texts do so. 2. You may feel like these aren't really definitions, and you're not entirely wrong. A rigor- ous definition of the natural numbers is an extremely tedious exercise in mathematical logic; famously, Russell and Whitehead feature the proposition that \1 + 1 = 2" on page 379 of Principia Mathematica. We will simply trust that everyone in this course understands how to count. The natural numbers have two very important properties. Fact 1.3 (The Well-Ordering Property). Every subset of the natural numbers has a least element. -
Cyclic Groups and Primitive Roots
Cyclic Groups and Primitive Roots Ryan C. Daileda Trinity University Number Theory Daileda Primitive Roots Cyclic Subgroups Let G be a group and let a ∈ G. The set hai = {am | m ∈ Z} is clearly closed under multiplication and inversion in G. hai is therefore a group in its own right, the cyclic subgroup generated by a. Our work last time immediately proves: Theorem 1 Let G be a group and let a ∈ G. If |a| is infinite, so is hai. If |a| = n ∈ N, then hai = {e, a, a2,..., an−1}, and these elements are all distinct. Daileda Primitive Roots Examples The (additive) subgroup of Z/20Z generated by 12 + 20Z is {12 + 20Z, 4 + 20Z, 16 + 20Z, 8 + 20Z, 0 + 20Z}, 20 which has 5 = (12,20) elements, as expected. The (multiplicative) subgroup of (Z/16Z)× generated by 3 + 16Z is {3 + 16Z, 9 + 16Z, 11 + 16Z, 1 + 16Z}, which has 4 = |3 + 16Z| elements. Daileda Primitive Roots Cyclic Groups Definition A group G is called cyclic if there is an a ∈ G so that G = hai. In this case we say that G is generated by a. Since |a| = hai , if G is finite we find that G is cyclic ⇔ G has an element of order |G|. Since the additive order of 1+ nZ in Z/nZ is exactly n, we conclude that Z/nZ (under addition) is always cyclic. Daileda Primitive Roots n Recall that the additive order of a + nZ in Z/nZ is (a,n) . Thus: The (additive) generators of Z/nZ are the elements of (Z/nZ)×. -
On Waring's Problem: Three
J. Brudern¨ and T. D. Wooley Nagoya Math. J. Vol. 163 (2001), 13{53 ON WARING'S PROBLEM: THREE CUBES AND A SIXTH POWER JOR¨ G BRUDERN¨ and TREVOR D. WOOLEY1 Abstract. We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers. 1. Introduction x It is widely expected that for any fixed positive integer k, all large natural numbers satisfying the necessary congruence conditions should be representable as the sum of three cubes and a kth power of natural numbers. Let νk(n) denote the number of representations of the positive integer n in this manner. Then a formal application of the circle method leads to the conjecture that the asymptotic formula 3 ν (n) Γ 4 S (n)n1=k k ∼ 3 k should hold, where q 1 4 a 3 3 3 k S (n) = q− e (x + x + x + x n) k q 1 2 3 4 − q=1 a=1 1 x1;:::;x4 q X (a;qX)=1 ≤ X ≤ denotes the singular series associated with the representation problem at hand. -
2Y, X > 0, Y ≥ 0 1. Scope We Consider the Number of Ways to Repr
THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x2 − 2y, x > 0, y ≥ 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2y +n = x2 for fixed positive n. The computational strategy is to count the solutions separately for even and odd exponents y, and to handle the odd case with modular residues for most cases of n. 1. Scope We consider the number of ways to represent n > 0 as (1) n = x2 − 2y; for pairs of the non-negative unknowns x and y. Equivalently we count the y such that (2) n + 2y = x2 are perfect squares. Remark 1. For n = 0 the number of solutions is infinite because each even value of y creates one solution. The n for which the count is nonzero end up in [8, A051204]. Remark 2. Negative n for which solutions exists are listed in [8, A051213] [4,2] . The solutions are counted separately for even y and odd y and added up. The total count is at most 4 [6]. 2. Even Exponents y Counting the squares x2 of the format (2) considering only even y is basically a matter of considering the values y = 0; 2; 4;::: in turn and checking explicitly each y2 + n against being a square. An upper limit to the y is determined as follows: • The y-values in the range 2y < n are all checked individually. • For larger y the values of 2y + n on the left hand side of (2) are represented in binary by some most significant bit contributed by 2y and|after a train of zero bits depending of how much larger 2y is than n|the trailing bits of n.