The Integer-Valued Polynomials on Lucas Numbers
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Fast Tabulation of Challenge Pseudoprimes Andrew Shallue and Jonathan Webster
THE OPEN BOOK SERIES 2 ANTS XIII Proceedings of the Thirteenth Algorithmic Number Theory Symposium Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster msp THE OPEN BOOK SERIES 2 (2019) Thirteenth Algorithmic Number Theory Symposium msp dx.doi.org/10.2140/obs.2019.2.411 Fast tabulation of challenge pseudoprimes Andrew Shallue and Jonathan Webster We provide a new algorithm for tabulating composite numbers which are pseudoprimes to both a Fermat test and a Lucas test. Our algorithm is optimized for parameter choices that minimize the occurrence of pseudoprimes, and for pseudoprimes with a fixed number of prime factors. Using this, we have confirmed that there are no PSW-challenge pseudoprimes with two or three prime factors up to 280. In the case where one is tabulating challenge pseudoprimes with a fixed number of prime factors, we prove our algorithm gives an unconditional asymptotic improvement over previous methods. 1. Introduction Pomerance, Selfridge, and Wagstaff famously offered $620 for a composite n that satisfies (1) 2n 1 1 .mod n/ so n is a base-2 Fermat pseudoprime, Á (2) .5 n/ 1 so n is not a square modulo 5, and j D (3) Fn 1 0 .mod n/ so n is a Fibonacci pseudoprime, C Á or to prove that no such n exists. We call composites that satisfy these conditions PSW-challenge pseudo- primes. In[PSW80] they credit R. Baillie with the discovery that combining a Fermat test with a Lucas test (with a certain specific parameter choice) makes for an especially effective primality test[BW80]. -
On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
mathematics Article On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function ŠtˇepánHubálovský 1,* and Eva Trojovská 2 1 Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic 2 Department of Mathematics, Faculty of Science, University of Hradec Králové, 50003 Hradec Králové, Czech Republic; [email protected] * Correspondence: [email protected] or [email protected]; Tel.: +420-49-333-2704 Received: 3 October 2020; Accepted: 14 October 2020; Published: 16 October 2020 Abstract: Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk ≡ 0 (mod n). In this paper, we shall find all positive solutions of the Diophantine equation z(j(n)) = n, where j is the Euler totient function. Keywords: Fibonacci numbers; order of appearance; Euler totient function; fixed points; Diophantine equations MSC: 11B39; 11DXX 1. Introduction Let (Fn)n≥0 be the sequence of Fibonacci numbers which is defined by 2nd order recurrence Fn+2 = Fn+1 + Fn, with initial conditions Fi = i, for i 2 f0, 1g. These numbers (together with the sequence of prime numbers) form a very important sequence in mathematics (mainly because its unexpectedly and often appearance in many branches of mathematics as well as in another disciplines). We refer the reader to [1–3] and their very extensive bibliography. We recall that an arithmetic function is any function f : Z>0 ! C (i.e., a complex-valued function which is defined for all positive integer). -
An Analysis of Primality Testing and Its Use in Cryptographic Applications
An Analysis of Primality Testing and Its Use in Cryptographic Applications Jake Massimo Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Information Security Royal Holloway, University of London 2020 Declaration These doctoral studies were conducted under the supervision of Prof. Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathe- matics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Jake Massimo April, 2020 2 Abstract Due to their fundamental utility within cryptography, prime numbers must be easy to both recognise and generate. For this, we depend upon primality testing. Both used as a tool to validate prime parameters, or as part of the algorithm used to generate random prime numbers, primality tests are found near universally within a cryptographer's tool-kit. In this thesis, we study in depth primality tests and their use in cryptographic applications. We first provide a systematic analysis of the implementation landscape of primality testing within cryptographic libraries and mathematical software. We then demon- strate how these tests perform under adversarial conditions, where the numbers being tested are not generated randomly, but instead by a possibly malicious party. We show that many of the libraries studied provide primality tests that are not pre- pared for testing on adversarial input, and therefore can declare composite numbers as being prime with a high probability. -
Triangular Numbers /, 3,6, 10, 15, ", Tn,'" »*"
TRIANGULAR NUMBERS V.E. HOGGATT, JR., and IVIARJORIE BICKWELL San Jose State University, San Jose, California 9111112 1. INTRODUCTION To Fibonacci is attributed the arithmetic triangle of odd numbers, in which the nth row has n entries, the cen- ter element is n* for even /?, and the row sum is n3. (See Stanley Bezuszka [11].) FIBONACCI'S TRIANGLE SUMS / 1 =:1 3 3 5 8 = 2s 7 9 11 27 = 33 13 15 17 19 64 = 4$ 21 23 25 27 29 125 = 5s We wish to derive some results here concerning the triangular numbers /, 3,6, 10, 15, ", Tn,'" »*". If one o b - serves how they are defined geometrically, 1 3 6 10 • - one easily sees that (1.1) Tn - 1+2+3 + .- +n = n(n±M and (1.2) • Tn+1 = Tn+(n+1) . By noticing that two adjacent arrays form a square, such as 3 + 6 = 9 '.'.?. we are led to 2 (1.3) n = Tn + Tn„7 , which can be verified using (1.1). This also provides an identity for triangular numbers in terms of subscripts which are also triangular numbers, T =T + T (1-4) n Tn Tn-1 • Since every odd number is the difference of two consecutive squares, it is informative to rewrite Fibonacci's tri- angle of odd numbers: 221 222 TRIANGULAR NUMBERS [OCT. FIBONACCI'S TRIANGLE SUMS f^-O2) Tf-T* (2* -I2) (32-22) Ti-Tf (42-32) (52-42) (62-52) Ti-Tl•2 (72-62) (82-72) (9*-82) (Kp-92) Tl-Tl Upon comparing with the first array, it would appear that the difference of the squares of two consecutive tri- angular numbers is a perfect cube. -
Primes and Primality Testing
Primes and Primality Testing A Technological/Historical Perspective Jennifer Ellis Department of Mathematics and Computer Science What is a prime number? A number p greater than one is prime if and only if the only divisors of p are 1 and p. Examples: 2, 3, 5, and 7 A few larger examples: 71887 524287 65537 2127 1 Primality Testing: Origins Eratosthenes: Developed “sieve” method 276-194 B.C. Nicknamed Beta – “second place” in many different academic disciplines Also made contributions to www-history.mcs.st- geometry, approximation of andrews.ac.uk/PictDisplay/Eratosthenes.html the Earth’s circumference Sieve of Eratosthenes 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Sieve of Eratosthenes We only need to “sieve” the multiples of numbers less than 10. Why? (10)(10)=100 (p)(q)<=100 Consider pq where p>10. Then for pq <=100, q must be less than 10. By sieving all the multiples of numbers less than 10 (here, multiples of q), we have removed all composite numbers less than 100. -
On the Discriminator of Lucas Sequences
Ann. Math. Québec (2019) 43:51–71 https://doi.org/10.1007/s40316-017-0097-7 On the discriminator of Lucas sequences Bernadette Faye1,2 · Florian Luca3,4,5 · Pieter Moree4 Received: 18 August 2017 / Accepted: 28 December 2017 / Published online: 12 February 2018 © The Author(s) 2018. This article is an open access publication Abstract We consider the family of Lucas sequences uniquely determined by Un+2(k) = (4k+2)Un+1(k)−Un(k), with initial values U0(k) = 0andU1(k) = 1andk ≥ 1anarbitrary integer. For any integer n ≥ 1 the discriminator function Dk(n) of Un(k) is defined as the smallest integer m such that U0(k), U1(k),...,Un−1(k) are pairwise incongruent modulo m. Numerical work of Shallit on Dk(n) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every k ≥ 1there is a constant nk such that Dk(n) has a simple characterization for every n ≥ nk. The case k = 1 turns out to be fundamentally different from the case k > 1. FRENCH ABSTRACT Pour un entier arbitraire k ≥ 1, on considère la famille de suites de Lucas déterminée de manière unique par la relation de récurrence Un+2(k) = (4k + 2)Un+1(k) − Un(k), et les valeurs initiales U0(k) = 0etU1(k) = 1. Pour tout entier n ≥ 1, la fonction discriminante Dk(n) de Un(k) est définie comme le plus petit m tel que U0(k), U1(k),...,Un−1(k) soient deux à deux non congruents modulo m. -
Arxiv:Math/0201082V1
2000]11A25, 13J05 THE RING OF ARITHMETICAL FUNCTIONS WITH UNITARY CONVOLUTION: DIVISORIAL AND TOPOLOGICAL PROPERTIES. JAN SNELLMAN Abstract. We study (A, +, ⊕), the ring of arithmetical functions with unitary convolution, giving an isomorphism between (A, +, ⊕) and a generalized power series ring on infinitely many variables, similar to the isomorphism of Cashwell- Everett[4] between the ring (A, +, ·) of arithmetical functions with Dirichlet convolution and the power series ring C[[x1,x2,x3,... ]] on countably many variables. We topologize it with respect to a natural norm, and shove that all ideals are quasi-finite. Some elementary results on factorization into atoms are obtained. We prove the existence of an abundance of non-associate regular non-units. 1. Introduction The ring of arithmetical functions with Dirichlet convolution, which we’ll denote by (A, +, ·), is the set of all functions N+ → C, where N+ denotes the positive integers. It is given the structure of a commutative C-algebra by component-wise addition and multiplication by scalars, and by the Dirichlet convolution f · g(k)= f(r)g(k/r). (1) Xr|k Then, the multiplicative unit is the function e1 with e1(1) = 1 and e1(k) = 0 for k> 1, and the additive unit is the zero function 0. Cashwell-Everett [4] showed that (A, +, ·) is a UFD using the isomorphism (A, +, ·) ≃ C[[x1, x2, x3,... ]], (2) where each xi corresponds to the function which is 1 on the i’th prime number, and 0 otherwise. Schwab and Silberberg [9] topologised (A, +, ·) by means of the norm 1 arXiv:math/0201082v1 [math.AC] 10 Jan 2002 |f| = (3) min { k f(k) 6=0 } They noted that this norm is an ultra-metric, and that ((A, +, ·), |·|) is a valued ring, i.e. -
And Its Properties on the Sequence Related to Lucas Numbers
Mathematica Aeterna, Vol. 2, 2012, no. 1, 63 - 75 On the sequence related to Lucas numbers and its properties Cennet Bolat Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Ahmet I˙pek Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Hasan Köse Department of Mathematics, Science Faculty, Selcuk University 42031, Konya,Turkey Abstract The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recur- rence relation. In this article, we study a new generalization {Lk,n}, with initial conditions Lk,0 = 2 and Lk,1 = 1, which is generated by the recurrence relation Lk,n = kLk,n−1 + Lk,n−2 for n ≥ 2, where k is integer number. Some well-known sequence are special case of this generalization. The Lucas sequence is a special case of {Lk,n} with k = 1. Modified Pell-Lucas sequence is {Lk,n} with k = 2. We produce an extended Binet’s formula for {Lk,n} and, thereby, identities such as Cassini’s, Catalan’s, d’Ocagne’s, etc. using matrix algebra. Moreover, we present sum formulas concerning this new generalization. Mathematics Subject Classi…cation: 11B39, 15A23. Keywords: Classic Fibonacci numbers; Classic Lucas numbers; k-Fibonacci numbers; k-Lucas numbers, Matrix algebra. 64 C. Bolat, A. Ipek and H. Kose 1 Introduction In recent years, many interesting properties of classic Fibonacci numbers, clas- sic Lucas numbers and their generalizations have been shown by researchers and applied to almost every …eld of science and art. -
Fixed Points of Certain Arithmetic Functions
FIXED POINTS OF CERTAIN ARITHMETIC FUNCTIONS WALTER E. BECK and RUDOLPH M. WAJAR University of Wisconsin, Whitewater, Wisconsin 53190 IWTRODUCTIOW Perfect, amicable and sociable numbers are fixed points of the arithemetic function L and its iterates, L (n) = a(n) - n, where o is the sum of divisor's function. Recently there have been investigations into functions differing from L by 1; i.e., functions L+, Z.,, defined by L + (n)= L(n)± 1. Jerrard and Temperley [1] studied the existence of fixed points of L+ and /._. Lai and Forbes [2] conducted a computer search for fixed points of (LJ . For earlier references to /._, see the bibliography in [2]. We consider the analogous situation using o*, the sum of unitary divisors function. Let Z.J, Lt, be arithmetic functions defined by L*+(n) = o*(n)-n±1. In § 1, we prove, using parity arguments, that L* has no fixed points. Fixed points of iterates of L* arise in sets where the number of elements in the set is equal to the power of/.* in question. In each such set there is at least one natural number n such that L*(n) > n. In § 2, we consider conditions n mustsatisfy to enjoy the inequality and how the inequality acts under multiplication. In particular if n is even, it is divisible by at least three primes; if odd, by five. If /? enjoys the inequality, any multiply by a relatively prime factor does so. There is a bound on the highest power of n that satisfies the inequality. Further if n does not enjoy the inequality, there are bounds on the prime powers multiplying n which will yield the inequality. -
On Some New Arithmetic Functions Involving Prime Divisors and Perfect Powers
On some new arithmetic functions involving prime divisors and perfect powers. Item Type Article Authors Bagdasar, Ovidiu; Tatt, Ralph-Joseph Citation Bagdasar, O., and Tatt, R. (2018) ‘On some new arithmetic functions involving prime divisors and perfect powers’, Electronic Notes in Discrete Mathematics, 70, pp.9-15. doi: 10.1016/ j.endm.2018.11.002 DOI 10.1016/j.endm.2018.11.002 Publisher Elsevier Journal Electronic Notes in Discrete Mathematics Download date 30/09/2021 07:19:18 Item License http://creativecommons.org/licenses/by/4.0/ Link to Item http://hdl.handle.net/10545/623232 On some new arithmetic functions involving prime divisors and perfect powers Ovidiu Bagdasar and Ralph Tatt 1,2 Department of Electronics, Computing and Mathematics University of Derby Kedleston Road, Derby, DE22 1GB, United Kingdom Abstract Integer division and perfect powers play a central role in numerous mathematical results, especially in number theory. Classical examples involve perfect squares like in Pythagora’s theorem, or higher perfect powers as the conjectures of Fermat (solved in 1994 by A. Wiles [8]) or Catalan (solved in 2002 by P. Mih˘ailescu [4]). The purpose of this paper is two-fold. First, we present some new integer sequences a(n), counting the positive integers smaller than n, having a maximal prime factor. We introduce an arithmetic function counting the number of perfect powers ij obtained for 1 ≤ i, j ≤ n. Along with some properties of this function, we present the sequence A303748, which was recently added to the Online Encyclopedia of Integer Sequences (OEIS) [5]. Finally, we discuss some other novel integer sequences. -
The Bivariate (Complex) Fibonacci and Lucas Polynomials: an Historical Investigation with the Maple's Help
Volume 9, Number 4, 2016 THE BIVARIATE (COMPLEX) FIBONACCI AND LUCAS POLYNOMIALS: AN HISTORICAL INVESTIGATION WITH THE MAPLE´S HELP Francisco Regis Vieira Alves, Paula Maria Machado Cruz Catarino Abstract: The current research around the Fibonacci´s and Lucas´ sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixthies. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it´s polinomial representations. Therefore, with the introduction of one or two variables, we begin to discuss the family of the Bivariate Lucas Polynomias (BLP) and the Bivariate Fibonacci Polynomials (BFP). On the other hand, since it´s representation requires enormous employment of a large algebraic notational system, we explore some particular properties in order to convince the reader about an inductive reasoning that produces a meaning and produces an environment of scientific and historical investigation supported by the technology. Finally, throughout the work we bring several figures that represent some examples of commands and algebraic operations with the CAS Maple that allow to compare properties of the Lucas´polynomials, taking as a reference the classic of Fibonacci´s model that still serves as inspiration for several current studies in Mathematics. Key words: Lucas Sequence, Fibonacci´s polynomials, Historical investigation, CAS Maple. 1. Introduction Undoubtedly, the Fibonacci sequence preserves a character of interest and, at the same time, of mystery, around the numerical properties of a sequence that is originated from a problem related to the infinite reproduction of pairs of rabbits. On the other hand, in several books of History of Mathematics in Brazil and in the other countries (Eves, 1969; Gullberg, 1997; Herz, 1998; Huntley, 1970; Vajda, 1989), we appreciate a naive that usually emphasizes eminently basic and trivial properties related to this sequence. -
On the Implementation of Pairing-Based Cryptosystems a Dissertation Submitted to the Department of Computer Science and the Comm
ON THE IMPLEMENTATION OF PAIRING-BASED CRYPTOSYSTEMS A DISSERTATION SUBMITTED TO THE DEPARTMENT OF COMPUTER SCIENCE AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Ben Lynn June 2007 c Copyright by Ben Lynn 2007 All Rights Reserved ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Dan Boneh Principal Advisor I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. John Mitchell I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Xavier Boyen Approved for the University Committee on Graduate Studies. iii Abstract Pairing-based cryptography has become a highly active research area. We define bilinear maps, or pairings, and show how they give rise to cryptosystems with new functionality. There is only one known mathematical setting where desirable pairings exist: hyperellip- tic curves. We focus on elliptic curves, which are the simplest case, and also the only curves used in practice. All existing implementations of pairing-based cryptosystems are built with elliptic curves. Accordingly, we provide a brief overview of elliptic curves, and functions known as the Tate and Weil pairings from which cryptographic pairings are derived. We describe several methods for obtaining curves that yield Tate and Weil pairings that are efficiently computable yet are still cryptographically secure.