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Pseudoprime Reductions of Elliptic Curves
Mathematical Proceedings of the Cambridge Philosophical Society VOL. 146 MAY 2009 PART 3 Math. Proc. Camb. Phil. Soc. (2009), 146, 513 c 2008 Cambridge Philosophical Society 513 doi:10.1017/S0305004108001758 Printed in the United Kingdom First published online 14 July 2008 Pseudoprime reductions of elliptic curves BY ALINA CARMEN COJOCARU Dept. of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045, U.S.A. and The Institute of Mathematics of the Romanian Academy, Bucharest, Romania. e-mail: [email protected] FLORIAN LUCA Instituto de Matematicas,´ Universidad Nacional Autonoma´ de Mexico,´ C.P. 58089, Morelia, Michoacan,´ Mexico.´ e-mail: [email protected] AND IGOR E. SHPARLINSKI Dept. of Computing, Macquarie University, Sydney, NSW 2109, Australia. e-mail: [email protected] (Received 10 April 2007; revised 25 April 2008) Abstract Let b 2 be an integer and let E/Q be a fixed elliptic curve. In this paper, we estimate the number of primes p x such that the number of points n E (p) on the reduction of E modulo p is a base b prime or pseudoprime. In particular, we improve previously known bounds which applied only to prime values of n E (p). 1. Introduction Let b 2 be an integer. Recall that a pseudoprime to base b is a composite positive integer m such that the congruence bm ≡ b (mod m) holds. The question of the distri- bution of pseudoprimes in certain sequences of positive integers has received some in- terest. For example, in [PoRo], van der Poorten and Rotkiewicz show that any arithmetic 514 A. -
Interprime 198 Yacht Marine for the Maintenance of Above Water Areas
Interprime 198 Yacht Marine For the maintenance of above water areas. PRODUCT DESCRIPTION A quick drying, one pack primer. Interprime 198 is surface tolerant, compatible with most substrates and can be overcoated with a wide range of finishes. PRODUCT INFORMATION Colour CPA097-White, CPA098-Grey, CPA099-Red Finish Matt (ISO 2813 : 1978) Specific Gravity 1.25 Volume Solids 41% average (ISO 3233 : 1998) Typical Shelf Life 1 yrs VOC (As Supplied) 506 g/lt Unit Size 5 Lt 20 Lt DRYING/OVERCOATING INFORMATION Drying 5ºC 15ºC 23ºC 35ºC Touch Dry [ISO] 3hrs 1.5hrs 60mins 30mins Hard Dry [ISO] 24hrs 10hrs 4hrs 2hrs Overcoating Substrate Temperature 5ºC 15ºC 23ºC 35ºC Overcoated By Min Max Min Max Min Max Min Max Interlac 665 6hrs 3days 4hrs 2.5days 2hrs 2days 60mins 24hrs Interprime 198 3hrs ext 2hrs ext 60mins ext 30mins ext Interthane 990 24hrs 7days 18hrs 7days 12hrs 7days 6hrs 3days Pre-Kote 24hrs 5days 18hrs 4days 12hrs 3days 8hrs 24hrs APPLICATION AND USE Preparation Use in accordance with the standard Global Yacht specifications. All surfaces to be coated should be clean, dry and free from contamination. High pressure fresh water wash or fresh water wash as appropriate, to remove all oil, grease, soluble contaminates and other detrimental foreign matter. Maintenance & Repair: Prepare area to be repaired to a minimum of St2 (ISO 8501-1:1998). Higher levels of surface preparation by abrasive blasting to Sa 2 (ISO 8501 - 1:1998) or hydroblasting to HB2M (International Hydroblasting Standards), will enhance product performance. Feather or chip back surrounding area to a sound edge. -
Number 73 Is the 37Th Odd Number
73 Seventy-Three LXXIII i ii iii iiiiiiiiii iiiiiiiii iiiiiiii iiiiiii iiiiiiii iiiiiiiii iiiiiiiiii iii ii i Analogous ordinal: seventy-third. The number 73 is the 37th odd number. Note the reversal here. The number 73 is the twenty-first prime number. The number 73 is the fifty-sixth deficient number. The number 73 is in the eighth twin-prime pair 71, 73. This is the last of the twin primes less than 100. No one knows if the list of twin primes ever ends. As a sum of four or fewer squares: 73 = 32 + 82 = 12 + 62 + 62 = 12 + 22 + 22 + 82 = 22 + 22 + 42 + 72 = 42 + 42 + 42 + 52. As a sum of nine or fewer cubes: 73 = 3 13 + 2 23 + 2 33 = 13 + 23 + 43. · · · As a difference of two squares: 73 = 372 362. The number 73 appears in two Pythagorean triples: [48, 55, 73] and [73, 2664, 2665]. Both are primitive, of course. As a sum of three odd primes: 73 = 3 + 3 + 67 = 3 + 11 + 59 = 3 + 17 + 53 = 3 + 23 + 47 = 3 + 29 + 41 = 5 + 7 + 61 = 5 + 31 + 37 = 7 + 7 + 59 = 7 + 13 + 53 = 7 + 19 + 47 = 7 + 23 + 43 = 7 + 29 + 37 = 11 + 19 + 43 = 11 + 31 + 31 = 13 + 13 + 47 = 13 + 17 + 43 = 13 + 19 + 41 = 13 + 23 + 37 = 13 + 29 + 31 = 17 + 19 + 37 = 19 + 23 + 31. The number 73 is the 21st prime number. It’s reversal, 37, is the 12th prime number. Notice that if you strip off the six triangular points from the 73-circle hexagram pictured above, you are left with a hexagon of 37 circles. -
Recent Progress in Additive Prime Number Theory
Introduction Arithmetic progressions Other linear patterns Recent progress in additive prime number theory Terence Tao University of California, Los Angeles Mahler Lecture Series Terence Tao Recent progress in additive prime number theory Introduction Arithmetic progressions Other linear patterns Additive prime number theory Additive prime number theory is the study of additive patterns in the prime numbers 2; 3; 5; 7;:::. Examples of additive patterns include twins p; p + 2, arithmetic progressions a; a + r;:::; a + (k − 1)r, and prime gaps pn+1 − pn. Many open problems regarding these patterns still remain, but there has been some recent progress in some directions. Terence Tao Recent progress in additive prime number theory Random models for the primes Introduction Sieve theory Arithmetic progressions Szemerédi’s theorem Other linear patterns Putting it together Long arithmetic progressions in the primes I’ll first discuss a theorem of Ben Green and myself from 2004: Theorem: The primes contain arbitrarily long arithmetic progressions. Terence Tao Recent progress in additive prime number theory Random models for the primes Introduction Sieve theory Arithmetic progressions Szemerédi’s theorem Other linear patterns Putting it together It was previously established by van der Corput (1929) that the primes contained infinitely many progressions of length three. In 1981, Heath-Brown showed that there are infinitely many progressions of length four, in which three elements are prime and the fourth is an almost prime (the product of at most -
A Member of the Order of Minims, Was a Philosopher, Mathematician, and Scientist Who Wrote on Music, Mechanics, and Optics
ON THE FORM OF MERSENNE NUMBERS PREDRAG TERZICH Abstract. We present a theorem concerning the form of Mersenne p numbers , Mp = 2 − 1 , where p > 3 . We also discuss a closed form expression which links prime numbers and natural logarithms . 1. Introduction. The friar Marin Mersenne (1588 − 1648), a member of the order of Minims, was a philosopher, mathematician, and scientist who wrote on music, mechanics, and optics . A Mersenne number is a number of p the form Mp = 2 − 1 where p is an integer , however many authors prefer to define a Mersenne number as a number of the above form but with p restricted to prime values . In this paper we shall use the second definition . Because the tests for Mersenne primes are relatively simple, the record largest known prime number has almost always been a Mersenne prime . 2. A theorem and closed form expression Theorem 2.1. Every Mersenne number Mp , (p > 3) is a number of the form 6 · k · p + 1 for some odd possitive integer k . Proof. In order to prove this we shall use following theorems : Theorem 2.2. The simultaneous congruences n ≡ n1 (mod m1) , n ≡ n2 (mod m2), .... ,n ≡ nk (mod mk) are only solvable when ni = nj (mod gcd(mi; mj)) , for all i and j . i 6= j . The solution is unique modulo lcm(m1; m2; ··· ; mk) . Proof. For proof see [1] Theorem 2.3. If p is a prime number, then for any integer a : ap ≡ a (mod p) . Date: January 4 , 2012. 1 2 PREDRAG TERZICH Proof. -
The Pseudoprimes to 25 • 109
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 151 JULY 1980, PAGES 1003-1026 The Pseudoprimes to 25 • 109 By Carl Pomerance, J. L. Selfridge and Samuel S. Wagstaff, Jr. Abstract. The odd composite n < 25 • 10 such that 2n_1 = 1 (mod n) have been determined and their distribution tabulated. We investigate the properties of three special types of pseudoprimes: Euler pseudoprimes, strong pseudoprimes, and Car- michael numbers. The theoretical upper bound and the heuristic lower bound due to Erdös for the counting function of the Carmichael numbers are both sharpened. Several new quick tests for primality are proposed, including some which combine pseudoprimes with Lucas sequences. 1. Introduction. According to Fermat's "Little Theorem", if p is prime and (a, p) = 1, then ap~1 = 1 (mod p). This theorem provides a "test" for primality which is very often correct: Given a large odd integer p, choose some a satisfying 1 <a <p - 1 and compute ap~1 (mod p). If ap~1 pi (mod p), then p is certainly composite. If ap~l = 1 (mod p), then p is probably prime. Odd composite numbers n for which (1) a"_1 = l (mod«) are called pseudoprimes to base a (psp(a)). (For simplicity, a can be any positive in- teger in this definition. We could let a be negative with little additional work. In the last 15 years, some authors have used pseudoprime (base a) to mean any number n > 1 satisfying (1), whether composite or prime.) It is well known that for each base a, there are infinitely many pseudoprimes to base a. -
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). -
Prime Numbers and Cryptography
the sunday tImes of malta maRCh 11, 2018 | 51 LIFE &WELLBEING SCIENCE Prime numbers and cryptography ALEXANDER FARRUGIA factored into its two primes by Benjamin Moody – that is a number having 155 digits! (By comparison, the number You are given a number n, and you 6,436,609 has only seven digits.) The first 50 prime numbers. are asked to find the two whole It took his desktop computer 73 numbers a and b, both greater days to achieve this feat. Later, than 1, such that a times b is equal during the same year, a 768-bit MYTH to n. If I give you the number 21, number, having 232 digits, was say, then you would tell me “that factored into its constituent DEBUNKED is 7 times 3”. Or, if I give you 55, primes by a team of 13 academ - you would say “that is 11 times 5”. ics – it took them two years to do Now suppose I ask you for the so! This is the largest number Is 1 a prime two whole numbers, both greater that has been factored to date. than 1, whose product (multipli - Nowadays, most websites number? cation) equals the number employing the RSA algorithm If this question was asked 6,436,609. This is a much harder use at least a 2,048-bit (617- before the start of the 20th problem, right? I invite you to try digit) number to encrypt data. century, one would have to work it out, before you read on. Some websites such as Facebook invariably received a ‘yes’. -
Sds Interprime 198 Grey International Paint
CPA098_D4 Safety Data Sheet INTERPRIME 198 GREY Sales Order: Sales Order Bulk Sales Reference No.: CPA098 SDS Revision Date: 02/28/2019 SDS Revision Number: D4-3 1. Identification of the preparation and company 1.1. Product identifier Product Identity INTERPRIME 198 GREY Bulk Sales Reference No. CPA098 1.2. Relevant identified uses of the substance or mixture and uses advised against Intended Use Paints and Coatings 1.3. Details of the supplier of the safety data sheet Company Name International Paint LLC Manufacturer: Akzo Nobel Coatings International Paint 6001 Antoine Drive Houston, Texas 77091 National Supplier: Akzo Nobel Coatings Ltd. 110 Woodbine Downs Blvd. Unit #4 Etobicoke, Ontario Canada M9W 5S6 +1 (800) 618-1010 Emergency CHEMTREC (800) 424-9300 International Paint (713) 682-1711 Customer Service International Paint LLC (800) 589-1267 Fax No. (800) 631-7481 2. Hazard identification of the product 2.1. Classification of the substance or mixture Flam. Liq. 3;H226 Flammable liquid and vapor. Skin Irrit. 2;H315 Causes skin irritation. Eye Irrit. 2;H319 Causes serious eye irritation. Skin Sens. 1;H317 May cause an allergic skin reaction. Carc. 2;H351 Suspected of causing cancer. Aquatic Chronic 3;H412 Harmful to aquatic life with long lasting effects. 2.2. Label elements Using the Toxicity Data listed in section 11 & 12 the product is labelled as follows. 1/10 CPA098_D4 Warning. H226 Flammable liquid and vapour. H315 Causes skin irritation. H317 May cause an allergic skin reaction. H319 Causes serious eye irritation. H351 Suspected of causing cancer. H412 Harmful to aquatic life with long lasting effects. -
Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11
q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2. -
WXML Final Report: Prime Spacings
WXML Final Report: Prime Spacings Jayadev Athreya, Barry Forman, Kristin DeVleming, Astrid Berge, Sara Ahmed Winter 2017 1 Introduction 1.1 The initial problem In mathematics, prime gaps (defined as r = pn − pn−1) are widely studied due to the potential insights they may reveal about the distribution of the primes, as well as methods they could reveal about obtaining large prime numbers. Our research of prime spacings focused on studying prime intervals, which th is defined as pn − pn−1 − 1, where pn is the n prime. Specifically, we looked at what we defined as prime-prime intervals, that is primes having pn − pn−1 − 1 = r where r is prime. We have also researched how often different values of r show up as prime gaps or prime intervals, and why this distribution of values of r occurs the way that we have experimentally observed it to. 1.2 New directions The idea of studying the primality of prime intervals had not been previously researched, so there were many directions to take from the initial proposed problem. We saw multiple areas that needed to be studied: most notably, the prevalence of prime-prime intervals in the set of all prime intervals, and the distribution of the values of prime intervals. 1 2 Progress 2.1 Computational The first step taken was to see how many prime intervals up to the first nth prime were prime. Our original thoughts were that the amount of prime- prime intervals would grow at a rate of π(π(x)), since the function π tells us how many primes there are up to some number x. -
ON PRIME CHAINS 3 Can Be Found in Many Elementary Number Theory Texts, for Example [8]
ON PRIME CHAINS DOUGLAS S. STONES Abstract. Let b be an odd integer such that b ≡ ±1 (mod 8) and let q be a prime with primitive root 2 such that q does not divide b. We show q−2 that if (pk)k=0 is a sequence of odd primes such that pk = 2pk−1 + b for all 1 ≤ k ≤ q − 2, then either (a) q divides p0 + b, (b) p0 = q or (c) p1 = q. λ−1 For integers a,b with a ≥ 1, a sequence of primes (pk)k=0 such that pk = apk−1+b for all 1 ≤ k ≤ λ − 1 is called a prime chain of length λ based on the pair (a,b). This follows the terminology of Lehmer [7]. The value of pk is given by k k (a − 1) (1) p = a p0 + b k (a − 1) for all 0 ≤ k ≤ λ − 1. For prime chains based on the pair (2, +1), Cunningham [2, p. 241] listed three prime chains of length 6 and identified some congruences satisfied by the primes within prime chains of length at least 4. Prime chains based on the pair (2, +1) are now called Cunningham chains of the first kind, which we will call C+1 chains, for short. Prime chains based on the pair (2, −1) are called Cunningham chains of the second kind, which we will call C−1 chains. We begin with the following theorem which has ramifications on the maximum length of a prime chain; it is a simple corollary of Fermat’s Little Theorem.