Introduction to Higher Mathematics Unit #4: Number Theory
Total Page:16
File Type:pdf, Size:1020Kb

Load more
Recommended publications
-
The History of the Primality of One—A Selection of Sources
THE HISTORY OF THE PRIMALITY OF ONE|A SELECTION OF SOURCES ANGELA REDDICK, YENG XIONG, AND CHRIS K. CALDWELLy This document, in an altered and updated form, is available online: https:// cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html. Please use that document instead. The table below is a selection of sources which address the question \is the number one a prime number?" Whether or not one is prime is simply a matter of definition, but definitions follow use, context and tradition. Modern usage dictates that the number one be called a unit and not a prime. We choose sources which made the author's view clear. This is often difficult because of language and typographical barriers (which, when possible, we tried to reproduce for the primary sources below so that the reader could better understand the context). It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal Disquisitiones Arithmeticae [45], but his statement of the fundamental theorem of arithmetic makes his stand clear. Some (see, for example, V. A. Lebesgue and G. H. Hardy below) seemed ambivalent (or allowed it to depend on the context?)1 The first column, titled `prime,' is yes when the author defined one to be prime. This is just a raw list of sources; for an evaluation of the history see our articles [17] and [110]. Any date before 1200 is an approximation. We would be glad to hear of significant additions or corrections to this list. Date: May 17, 2016. 2010 Mathematics Subject Classification. -
The First 50 Million Prime Numbers*
7 sin ~F(s) The First 50 Million I - s ; sin -- Prime Numbers* s Don Zagier for the algebraist it is "the characteristic of a finite field" or "a point in Spec ~" or "a non-archimedean valuation"; To my parents the combinatorist defines the prime num- bers inductively by the recursion (I) 1 Pn+1 = [I - log2( ~ + n (-1)r )] r~1 ISil<'~'<irSn 2 pi1"''pir - I ([x] = biggest integer S x); and, finally, the logicians have recent- iv been defininq the primes as the posi- tive values of the polynomial (2) F(a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q, r,s,t,u,v,w,x,y,z) = [k + 2] [1 - (wz+h+j-q) 2 - (2n+p+q+z-e) 2 - (a2y2-y2+1-x2)2 - ({e4+2e3}{a+1}2+1-o2) 2 - (16{k+1}3{k+2}{n+1}2+1-f2)2 - ({(a+u~-u2a) 2-1}{n+4dy}2+1-{x+cu}2) 2 - (ai+k+1-l-i) 2 - I would like to tell you today about a ({gk+2g+k+1}{h+j}+h-z) 2 - subject which, although I have not worked (16r2y~{a2-1}+1-u2) 2 - in it myself, has always extraordinarily (p-m+l{a-n-1}+b{2an+2a-n2-2n-2}) 2 - captivated me, and which has fascinated (z-pm+pla-p21+t{2ap-p2-1}) 2 - mathematicians from the earliest times (qTx+y{a-p-1}+s{2ap+2a-p2-2~-2})2 - until the present - namely, the question (a~12-12+1-m2) 2 - (n+l+v-y) ]. -
THE DISTRIBUTION of PRIME NUMBERS Andrew Granville and K
THE DISTRIBUTION OF PRIME NUMBERS Andrew Granville and K. Soundararajan 0. Preface: Seduced by zeros 0.1. The prime number theorem, a brief history. A prime number is a positive integer which has no positive integer factors other than 1 and itself. It is difficult to determine directly from this definition whether there are many primes, indeed whether there are infinitely many. Euclid described in his Elements, an ancient Greek proof that there are infinitely many primes, a proof by contradiction, that today highlights for us the depth of abstract thinking in antiquity. So the next question is to quantify how many primes there are up to a given point. By studying tables of primes, Gauss understood, as a boy of 15 or 16 (in 1792 or 1 1793), that the primes occur with density log x at around x. In other words Z X 1 x dt π(x) := #{primes ≤ x} is approximately ≈ Li(x) where Li(x) := . log n log t n≤x 2 This leads to the conjecture that π(x)/Li(x) → 1 as x → ∞, which is hard to interpret x since Li(x) is not such a natural function. One can show that Li(x)/ log x → 1 as x → ∞, x so we can rephrase our conjecture as π(x)/ log x → 1 as x → ∞, or in less cumbersome notation that x (0.1.1) π(x) ∼ . log x It is not easy to find a strategy to prove such a result. Since primes are those integers with no prime factors less than√ or equal to their square-root, one obvious approach to counting the number of primes√ in ( x, x] is to try to estimate the number of integers up to x, with no prime factors ≤ x. -
Survey Report
YouGov / The Sun Survey Results Sample Size: 1923 GB Adults Fieldwork: 31st August - 1st September 2010 Voting intention Gender Age Social grade Region Lib Rest of Midlands Total Con Lab Male Female 18-24 25-39 40-59 60+ ABC1 C2DE London North Scotland Dem South / Wales Weighted Sample 1923 635 557 179 938 985 232 491 661 539 1079 814 247 625 414 470 167 Unweighted Sample 1923 622 545 194 916 1007 124 505 779 515 1244 649 239 661 370 422 231 % %%% % % %%%%% % % % % % % 10 - 11 31 Aug - May 1 Sept 2007 2010 Former Prime Minister Tony Blair is due to publish his memoirs Thinking back to his time in office, do you think Tony Blair...? Was a very good Prime Minister and achieved many 11 10 4 22 7 11 8 5 11 11 8 9 11 12 8 11 9 10 successes Was a fairly good Prime Minister - his successes 38 37 23 58 42 40 35 38 43 39 29 38 35 35 32 41 43 32 outnumbered his failures Was a fairlyyp poor Prime Minister - his failures outnumbered 28 21 28 11 27 22 20 21 21 19 22 23 19 17 25 18 18 24 his successes Was a very poor Prime Minister and can be blamed for many 18 25 42 7 22 23 27 14 15 26 39 25 26 28 27 23 24 26 failures Don't know 5 7 3 2 3 4 10 21 10 4 2 5 10 9 8 6 6 8 Since the second world war, six Prime Ministers have served for five years or longer. -
Package 'Numbers'
Package ‘numbers’ May 14, 2021 Type Package Title Number-Theoretic Functions Version 0.8-2 Date 2021-05-13 Author Hans Werner Borchers Maintainer Hans W. Borchers <[email protected]> Depends R (>= 3.1.0) Suggests gmp (>= 0.5-1) Description Provides number-theoretic functions for factorization, prime numbers, twin primes, primitive roots, modular logarithm and inverses, extended GCD, Farey series and continuous fractions. Includes Legendre and Jacobi symbols, some divisor functions, Euler's Phi function, etc. License GPL (>= 3) NeedsCompilation no Repository CRAN Date/Publication 2021-05-14 18:40:06 UTC R topics documented: numbers-package . .3 agm .............................................5 bell .............................................7 Bernoulli numbers . .7 Carmichael numbers . .9 catalan . 10 cf2num . 11 chinese remainder theorem . 12 collatz . 13 contFrac . 14 coprime . 15 div.............................................. 16 1 2 R topics documented: divisors . 17 dropletPi . 18 egyptian_complete . 19 egyptian_methods . 20 eulersPhi . 21 extGCD . 22 Farey Numbers . 23 fibonacci . 24 GCD, LCM . 25 Hermite normal form . 27 iNthroot . 28 isIntpower . 29 isNatural . 30 isPrime . 31 isPrimroot . 32 legendre_sym . 32 mersenne . 33 miller_rabin . 34 mod............................................. 35 modinv, modsqrt . 37 modlin . 38 modlog . 38 modpower . 39 moebius . 41 necklace . 42 nextPrime . 43 omega............................................ 44 ordpn . 45 Pascal triangle . 45 previousPrime . 46 primeFactors . 47 Primes -
Sparse Interpolation Over Finite Fields Via Low-Order Roots of Unity
Sparse interpolation over finite fields via low-order roots of unity Andrew Arnold Mark Giesbrecht Cheriton School of Computer Science Cheriton School of Computer Science University of Waterloo University of Waterloo [email protected] [email protected] Daniel S. Roche Computer Science Department United States Naval Academy [email protected] August 21, 2018 Abstract We present a new Monte Carlo algorithm for the interpolation of a straight-line program as a sparse polynomial f over an arbitrary finite field of size q. We assume a priori bounds D and T are given on the degree and number of terms of f. The approach presented in this paper is a hybrid of the diversified and recursive interpolation algorithms, the two previous fastest known probabilistic methods for this problem. By making effective use of the information contained in the coefficients themselves, this new algorithm improves on the bit complexity of previous methods by a “soft-Oh” factor of T , log D, or log q. 1 Introduction arXiv:1401.4744v2 [cs.SC] 2 May 2014 Let Fq be a finite field of size q and consider a “sparse” polynomial t ei F f = i=1 ciz ∈ q[z], (1.1) P where the c1,...,ct ∈ Fq are nonzero and the exponents e1,...,et ∈ Z≥0 are distinct. Suppose f is provided as a straight-line program, a simple branch-free program which evaluates f at any point (see below for a formal definition). Suppose also that we are given an upper bound D on the degree of f and an upper bound T on the number of non-zero terms t of f. -
Numbers 1 to 100
Numbers 1 to 100 PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 30 Nov 2010 02:36:24 UTC Contents Articles −1 (number) 1 0 (number) 3 1 (number) 12 2 (number) 17 3 (number) 23 4 (number) 32 5 (number) 42 6 (number) 50 7 (number) 58 8 (number) 73 9 (number) 77 10 (number) 82 11 (number) 88 12 (number) 94 13 (number) 102 14 (number) 107 15 (number) 111 16 (number) 114 17 (number) 118 18 (number) 124 19 (number) 127 20 (number) 132 21 (number) 136 22 (number) 140 23 (number) 144 24 (number) 148 25 (number) 152 26 (number) 155 27 (number) 158 28 (number) 162 29 (number) 165 30 (number) 168 31 (number) 172 32 (number) 175 33 (number) 179 34 (number) 182 35 (number) 185 36 (number) 188 37 (number) 191 38 (number) 193 39 (number) 196 40 (number) 199 41 (number) 204 42 (number) 207 43 (number) 214 44 (number) 217 45 (number) 220 46 (number) 222 47 (number) 225 48 (number) 229 49 (number) 232 50 (number) 235 51 (number) 238 52 (number) 241 53 (number) 243 54 (number) 246 55 (number) 248 56 (number) 251 57 (number) 255 58 (number) 258 59 (number) 260 60 (number) 263 61 (number) 267 62 (number) 270 63 (number) 272 64 (number) 274 66 (number) 277 67 (number) 280 68 (number) 282 69 (number) 284 70 (number) 286 71 (number) 289 72 (number) 292 73 (number) 296 74 (number) 298 75 (number) 301 77 (number) 302 78 (number) 305 79 (number) 307 80 (number) 309 81 (number) 311 82 (number) 313 83 (number) 315 84 (number) 318 85 (number) 320 86 (number) 323 87 (number) 326 88 (number) -
Heuristics on Class Groups: Some Good Primes Are Not Too Good
MATHEMATICSof computation VOLUME63, NUMBER207 JULY 1994, PAGES 329-334 HEURISTICS ON CLASS GROUPS: SOME GOOD PRIMES ARE NOT TOO GOOD HENRI COHEN AND JACQUESMARTINET Abstract. We correct some too optimistic predictions in an earlier paper of ours. 1 When generalizing to arbitrary relative extensions of number fields the Cohen and Lenstra heuristics of [1], it appeared that the natural hypothesis, namely not to include the p-components of the class groups for those p which divide the degree of a Galois closure, could be somewhat weakened, and we were led to the notion of a ""good prime" (cf. §2 below). Primes which do not divide the degree of the Galois closure are good, and we added some more primes to this list. In particular, 2 was considered to be a good prime for all cubic fields, though it divides the degree of the Galois closure when the field is not Galois. It was clear to us as soon as we began our work that these additional good primes could produce special difficulties [2, §3, p. 136]. However, the recent computations of G. Fung and H. Williams on nonreal cubic fields showed clearly that our recipes cannot be applied in a naive manner to the extra good primes. 2 Recall briefly some of the basic ideas of our heuristics. Let Ko be a fixed number field. We first consider (within a given algebraic closure of Q containing Ko ) a set of finite Galois extensions K/Ko with a given Galois group T (up to isomorphism) and for which the conjugacy classes of the infinite Frobenius substitutions attached to the real places of K0 are given. -
About the Author and This Text
1 PHUN WITH DIK AND JAYN BREAKING RSA CODES FOR FUN AND PROFIT A supplement to the text CALCULATE PRIMES By James M. McCanney, M.S. RSA-2048 has 617 decimal digits (2,048 bits). It is the largest of the RSA numbers and carried the largest cash prize for its factorization, US$200,000. The largest factored RSA number is 768 bits long (232 decimal digits). According to the RSA Corporation the RSA-2048 may not be factorable for many years to come, unless considerable advances are made in integer factorization or computational power in the near future Your assignment after reading this book (using the Generator Function of the book Calculate Primes) is to find the two prime factors of RSA-2048 (listed below) and mail it to the RSA Corporation explaining that you used the work of Professor James McCanney. RSA-2048 = 25195908475657893494027183240048398571429282126204032027777 13783604366202070759555626401852588078440691829064124951508 21892985591491761845028084891200728449926873928072877767359 71418347270261896375014971824691165077613379859095700097330 45974880842840179742910064245869181719511874612151517265463 22822168699875491824224336372590851418654620435767984233871 84774447920739934236584823824281198163815010674810451660377 30605620161967625613384414360383390441495263443219011465754 44541784240209246165157233507787077498171257724679629263863 56373289912154831438167899885040445364023527381951378636564 391212010397122822120720357 jmccanneyscience.com press - ISBN 978-0-9828520-3-3 $13.95US NOT FOR RESALE Copyrights 2006, 2007, 2011, 2014 -
Gambling, Saving, and Lumpy Liquidity Needs Sylvan Herskowitz Online Appendix A. Additional Figures and Tables B. Sports Betting
Gambling, Saving, and Lumpy Liquidity Needs Sylvan Herskowitz Online Appendix A. Additional Figures and Tables B. Sports Betting Details B.1 Odds, Payouts, and Betting Structure B.2 Estimating the Rate of Return B.3 Characterizing Weekly Betting Profiles C. Model and Comparative Statics C.1 Demand for Gambles C.2 Increased Valuation of a Lumpy Expenditure C.3 Demand for Saving C.4 Changes in Saving Ability D. Contrasting Saving and Betting D.1 Balancing Patience and Return on Saving D.2 Estimating Return on Saving E. Additional Field Targeting Protocols 1 Appendix A: Additional Tables and Figures Figure A.1: Indirect Utility with Lumpy Good and Demand for Gambles 푉 푢 푌 − 푃 + 휂 Purchase No Purchase 푢 푌 휂 푃 푌∗ 푌 (a) Indirect utility with a lumpy good V V C E F 퐵 − 퐻 = 휎 = D + D 푚푖푛 퐵 + 푊 A + 푌 푌 ∗ ∗ 퐻 � − � � (b) Fair Gambles (c) Unfair Gambles Notes: Panel (a) shows indirect utility from income. For income levels Y > Y ∗, people will pay a price P to consume L with a utility payoff of η. Maximized utility is determined by income endowment and will be the envelope of the two pieces of the utility function. Panel (b) shows that someone with income level Y˜ will demand a fair gamble that risks reducing his income by B∗ for a chance to win W ∗ with a likelihood of winning, σ. Expected utility from the gamble is at E. Panel (c) shows that there is also demand for unfair gambles with the same loss and win amounts but win likelihood as low as σmin. -
ON the SELBERG-ERD˝OS PROOF of the PRIME NUMBER THEOREM 1. Introduction the Prime Numbers, and Specifically Their Enumerating F
ON THE SELBERG-ERDOS} PROOF OF THE PRIME NUMBER THEOREM ASHVIN A. SWAMINATHAN Abstract. In this article, we discuss the first elementary proof, due to Selberg and Erd}os, of the Prime Number Theorem. In particular, we begin with a presentation of the Selberg symmetry formula and proceed to give a detailed account of the proof as published by Erd}osin 1949. 1. Introduction The prime numbers, and specifically their enumerating function π defined by π(x) = #fprimes p : p ≤ xg; have been a source of fascination to mathematicians for thousands of years. As early as 300 BC, Euclid discussed a simple proof of the infinitude of primes (i.e. the unboundedness of π) in his famed Elements. As late as the 1700s, Euler discovered another, more informative proof that there are infinitely many primes: in fact, he showed that the primes are so frequent among the integers that the series X 1 p prime p diverges. Drawing on Euler's methods and employing inputs from complex analy- sis, like the study of Dirichlet series and Landau's Theorem, Dirichlet proved that there are infinitely many primes in arithmetic progressions. Such tools, and in particular the complex analytic properties of the Riemann ζ-function, were funda- mental in establishing the first asymptotic for π(x). As an adolescent in the late R x 1700s, Gauss had correctly posited that π(x) ∼ x= log x ∼ Li(x) := 2 dt= log t, but it took a century for his conjecture to be proven. In the 1840s via elementary methods, Chebyshev explicitly computed constants c1 < 1 < c2, both close to 1 in value, such that π(x) (1) c < < c ; 1 x= log x 2 a result that provided a partial validation of Gauss' conjecture. -
1. Company Introduction 2. Factory Facilities 3. License 4. Products 5
0 - List - 1. Company introduction 2. Factory facilities 3. License 4. products 5. Control system 6. Main reference 1 1. Company Introduction Summarize company/establishment WORLD E&C Co., Ltd / March 20, 1999 • Absorption Chiller : Double life hot water, Single effect hot water, Steam fired absorption chiller, Heat Pump Item • Absorption chiller and heater : Gas/Oil, Exhaust gas, (Integrated type, Multi type) • Screw chiller, Condensing unit • Air Handling Unit : Exhaust heat recovery type, standard type • Total asset: USD 12 Million(2017) Finance • Annual sale : USD 18 Million(2017) Plottage 10,500㎡ • Factory Factory 4,500㎡ Factory scale (Hwasung-si) Office/R&D center 750㎡ • Anyang branch 165㎡ 662, Seokpo-ri, Jangan-myeon Hwaseong-si, Kyeonggi • Head office / factory Province, Republic of Korea Address • Sales/CS 1207, Digital empire B, 906-4, Gwanyang-dong, Dongan-gu, (Anyang branch) Anyang-si, Kyeonggi Province, Republic of Korea Homepage http://www.worldenc.com/ 2 Company History 2009 Development of exhaust gas absorption chiller-heater 2017 Initiate national project of development of Heat pump and transformer 2007 Registration of patent (Double Lift Hot water Driven Absorption Chiller) Initiate national project of development of Adsorption Chiller 2006 Development of Double lift Hot water Absorption Chiller 2016 Certificate of Designation of Excellent Product by Public Procurement Development of Single Effect Hot water Driven Absorption Chiller Service Registration of Direct Fired Absorption Chiller & Heater in Certificate INNO BIZ by SMBA Korea Public Procurement Agency Certificate of green technology 2005 Registration license for making specific facility, Certificate of venture company (Double Lift Hot water Driven Absorption Chiller) Establish research affiliated with World E&C Co., Ltd.