LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY


 Charlotte Gordon
 1 years ago
 Views:
Transcription
1 LECTURE 9: QUADRATIC RESIDUES AND THE LAW OF QUADRATIC RECIPROCITY 1. Bsic roerties of qudrtic residues We now investigte residues with secil roerties of lgebric tye. Definition 1.1. (i) When (, m) 1 nd x n (mod m) hs solution, then we sy tht is n nth ower residue modulo m. (ii) When (, m) 1, we sy tht is qudrtic residue modulo m rovided tht the congruence x 2 (mod m) is soluble. If the ltter congruence is insoluble, then we sy tht is qudrtic nonresidue modulo m. Theorem 1.2. Suose tht is rime number nd (, ) 1. Then the congruence x n (mod ) is soluble if nd only if 1 (n, 1) 1 (mod ). Proof. Let g be rimitive root modulo. Then for some nturl number r one hs g r (mod ). If 1 (n, 1) 1 (mod ), then g r( 1) (n, 1) 1 (mod ). But since g is rimitive, the ltter congruence cn hold only when ( 1) r( 1) (n, 1), whence (n, 1) r. But by the Eucliden Algorithm, there exist integers u nd v with nu + ( 1)v (n, 1), so on writing r k(n, 1), we obtin g k(n, 1) (g ku ) n (g 1 ) kv (g ku ) n (mod ). Thus is indeed n nth ower residue under these circumstnces. On the other hnd, if the congruence x n (mod ) is soluble, then 1 (n, 1) (x 1 ) n/(n, 1) 1 (mod ), on mking use of Fermt s Little Theorem. This comletes the roof of the theorem. Exmle 1.3. Determine whether or not 3 is 4th ower residue modulo 17. Observe tht on mking use of Theorem 1.2, the congruence x 4 3 (mod 17) is soluble if nd only if 3 16/4 1 (mod 17), tht is, if 81 1 (mod 17). Since this congruence is not stisfied, one finds tht 3 is not 4th ower residue modulo 17. 1
2 2 LECTURE 9 Definition ( ) 1.4. When is n odd rime number, define the Legendre symbol by +1, when is qudrtic residue modulo, 1, when is qudrtic nonresidue modulo, 0, when. Theorem 1.5 (Euler s criterion). When is n odd rime, one hs ( 1)/2 (mod ). Proof. If ( 1)/2 1 (mod ), then the desired conclusion is n immedite consequence of Theorem 1.2. The conclusion is lso immedite when. It remins to consider the sitution in which ( 1)/2 1 (mod ). Let be n integer with (, ) 1, write r ( 1)/2, nd note tht in view of Fermt s Little Theorem, one hs r (mod ), whence r ±1 (mod ). Then if r 1 (mod ), one necessrily hs r 1 (mod ). Thus, in the sitution in which ( 1)/2 1 (mod ), wherein Theorem 1.2 estblishes tht is qudrtic nonresidue modulo, one hs ( 1)/2 1 (mod ), nd so the desired conclusion follows once gin. This comletes the roof of the theorem. Theorem 1.6. Let be n odd rime number. Then (i) for ll integers nd b, one hs ( ) ( ) ( ) b b ; (ii) whenever b (mod ), one hs (iii) whenever (, ) 1, one hs 2 1 nd (iv) one hs ( ) 1 1 nd ( ) b ; 2 b ( ) b ; ( ) 1 ( 1) ( 1)/2. Proof. These conclusions re essentilly immedite from Theorem 1.5. exmle, the ltter theorem shows tht ( ) ( ) ( ) b b (b) ( 1)/2 ( 1)/2 b ( 1)/2 (mod ), nd so the conclusion of rt (i) of the theorem follows on noting tht since is odd, one cnnot hve 1 1 (mod ). Prts (ii) nd (iv) re trivil from the lst observtion, nd rt (iii) follows from Fermt s Little Theorem. For
3 LECTURE 9 3 Note: ( The ) number of solutions of the congruence x 2 (mod ) is given by 1 +. For when (, ) 1 nd the congruence is soluble, one hs two distinct solutions nd In the corresonding cse in which the congruence is insoluble, one hs ( 1) 0. When (, ) > 1, ( one ) the other hnd, one hs the single solution x 0 (mod ), nd then The bove observtion rovides mens of nlysing the solubility of qudrtic equtions. For if (, ) 1 nd > 2, then the congruence x 2 +bx+c 0 (mod ) is soluble if nd only if (2x + b) 2 b 2 4c (mod ) is soluble, tht is, if nd only if either b 2 4c 0 (mod ), or else ( ) b 2 4c 1. The number of solutions of the congruence is therefore recisely ( ) b 2 4c 1 +. ( ) It is cler from the multilictive roerty of tht it suffices now to ( ) ( ) ( ) q 2 comute for odd rime numbers q nd in order to clculte in generl. 2. The lw of qudrtic recirocity We now come to one of the most beutiful results of our course the Lw of Qudrtic Recirocity, which Guss clled the ureum theorem ( golden theorem ). Euler ws the first to mke conjectures equivlent to Qudrtic Recirocity, but he ws unble to rove it. Legendre lso worked on this roblem very seriously nd develoed mny vluble ides, in rticulr, he lso introduced the Legendre symbol. Finlly, Guss gve comlete roof of the Lw of Qudrtic Recirocity in 1797, when he ws 19. Now there re over 200 different roofs of this fundmentl result. Theorem 2.1 (Lw of Qudrtic Recirocity; Guss). Let nd q be distinct odd rime numbers. Then ( ) ( ) q ( 1) ( 1)(q 1)/4. q
4 4 LECTURE 9 Rewriting the exression on the right hnd side of the lst eqution in the she ( ) ( ) q ( 1) 1 2 ( 1) 1 2 (q 1), q ( ) ( ) q we see tht unless nd q re both congruent to 3 modulo 4. q We give the roof of qudrtic recirocity which is due to Eisenstein. It is bsed on the following wy to comute the Legendre symbol: Lemm 2.2 (Eisenstein). For n odd rime nd (, ) 1, ( 1) ( 1)/2 k1 2k/. Proof. Let E {2, 4,..., 1}. For every x E, we write x x/ + r x, 0 r x <. We observe tht ech of the numbers ( 1) rx r x is congruent to n element of E. This is cler when r x is even, nd when r x is odd, ( 1) rx r x r x (mod ) where r x E. We lso clim tht if ( 1) rx r x ( 1) ry r y (mod ), then x y. Indeed, if r x r y (mod ), then x y (mod ), nd it follows tht x y (mod ). If r x r y (mod ), then x y (mod ), nd x y (mod ), nd (x+y), but x+y 2( 1), so tht this is imossible. Hence, we conclude tht nd x E x x E On the other hnd, ( Since {( 1) rx r x (mod ) : x E} E, ( 1) rx r x ( 1) x E rx r x (mod ). r x x ( 1)/2 x E x E x E ) ( 1)/2 (mod ), we deduce tht ( 1) x E rx. x E x (mod ). Finlly, we observe tht r x x/ (mod 2). This imlies the lemm. ( ) ( ) q Proof of Qudrtic Recirocity. We shll use the formuls for nd q rovided by the Einsenstein Lemm. The min ide of the roof is tht the sums ( 1)/2 (q 1)/2 2kq/ nd 2k/q k1 cn be interreted geometriclly. k1
5 LECTURE 9 5 We my think bout 2kq/ s the number of oints (x, y) with x 2k nd y equl to ositive integer t most 2kq/. Then the sum ( 1)/2 k1 2kq/ is equl to the number of integrl oints with even xcoordinte contined in the interior of tringle ABD. Note tht there re no integrl oints on the line AB (why?). We note tht the number of integrl oints contined in the interior of rectngle AF BD nd lying on fixed integrl verticl line is equl to q 1, thus, even. This imlies tht the number of integrl oints in KHBD with even xcoordinte is equl modulo 2 to the number of integrl oints in HJB with even xcoordinte. We lso observe tht the trnsformtion (x, y) ( x, q y) send the integrl oints with even xcoordintes contined in HJB to the integrl oints with odd xcoordintes contined in AHK. Finlly, we conclude tht he number of integrl oints with even xcoordinte contined in the interior of tringle ABD is congruent modulo 2 to the sum of the integrl oints with odd xcoordintes contined in AHK lus the integrl oints with even xcoordintes contined in AHK, nmely, it is recisely the number of integrl oints contined in AHK. We obtin tht ( 1)/2 k1 2kq/ v 1 (mod 2), where v 1 is the number of integrl oints contined in AHK. The sme rgument gives tht (q 1)/2 k1 2k/q v 2 (mod 2), where v 2 is the number of integrl oints contined in ALH. In view of Eisenstein s Lemm, ( ) ( ) q ( 1) v 1+v 2, q
6 6 LECTURE 9 but v 1 + v 2 is recisely the number of integrl oints in ALHK. Hence, v 1 + v 2 1 q This comletes the roof. Theorem 2.3. For ny odd rime, ( ) 2 ( 1) (2 1)/8. Proof. We use tht ( 1)/2 k1 When 8m ± 1, then ( 1)/2 k1 4k/ {k N : /4 < k < /2} /2 /4. 4k/ 4m ± 1/2 2m ± 1/4 0 ( 2 1)/8 (mod 2). When 8m ± 3, then ( 1)/2 k1 4k/ 4m ± 3/2 2m ± 3/4 2m ± 1 1 ( 2 1)/8 (mod 2). Exmle 2.4. Determine the vlue of ( ) 3. By Qudrtic Recirocity we hve ( ) 3 ( ) ( 1) (3 1)( 1)/4 ( 1) ( 1)/2, 3 nd by Euler s criterion, on the other hnd, ( ) 1 ( 1) ( 1)/2. Thus we see tht ( ) 3 ( 1 ) ( ) 3 ( ) ( ( 1) ( 1)/2 ( 1) ( 1)/2. 3 3) But ( ) 1 ( 1, when 1 (mod 3), 3 ( ) 3) 2 1, when 2 (mod 3). 3 Thus we deduce tht ( ) { 3 1, when 1 (mod 3), 1, when 2 (mod 3).
7 LECTURE 9 7 One cn use this evlution to show tht the only ossible rime divisors of x 2 + 3, for integrl vlues of x, re 3 nd rimes with 1 (mod 3). From here, n rgument similr to tht due to Euclid shows tht there re infinitely mny rimes congruent to 1 modulo 3. ( ) 21 Exmle 2.5. Determine the vlue of. 71 Alying the multilictive roerty of the Legendre symbol, followed by qudrtic recirocity, one finds tht ( ) ( ) ( ) ( ) ( ) ( 1) (71 1)(3 1)/4+(71 1)(7 1)/ ( ) ( ) ( ) ( ) ( ) ( ) 21 So 1, nd hence 21 is not qudrtic residue modulo
MATH 118 PROBLEM SET 6
MATH 118 PROBLEM SET 6 WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER Section 1 #16: Show tht if is qudrtic residue modulo m, nd b 1 (mod m, then b is lso qudrtic residue Then rove tht the roduct of the
More informationSolutions to Exam 1. Problem 1. a) State Fermat s Little Theorem and Euler s Theorem. b) Let m, n be relatively prime positive integers.
Solutions to Exam 1 Problem 1. a) State Fermat s Little Theorem and Euler s Theorem. b) Let m, n be relatively rime ositive integers. Prove that m φ(n) + n φ(m) 1 (mod mn). c) Find the remainder of 1 008
More informationMATH 324 Elementary Number Theory Solutions to Practice Problems for Final Examination Monday August 8, 2005
MATH 324 Elementary Number Theory Solutions to Practice Problems for Final Examination Monday August 8, 2005 Deartment of Mathematical and Statistical Sciences University of Alberta Question 1. Find integers
More informationExam 1 7 = = 49 2 ( ) = = 7 ( ) =
Exam 1 Problem 1. a) Define gcd(a, b). Using Euclid s algorithm comute gcd(889, 168). Then find x, y Z such that gcd(889, 168) = x 889 + y 168 (check your answer!). b) Let a be an integer. Prove that gcd(3a
More informationNUMBER THEORY Amin Witno
WON Series in Discrete Mthemtics nd Modern Algebr Volume 2 NUMBER THEORY Amin Witno Prefce Written t Phildelphi University, Jordn for Mth 313, these notes 1 were used first time in the Fll 2005 semester.
More informationIs 1 a Square Modulo p? Is 2?
Chater 21 Is 1 a Square Modulo? Is 2? In the revious chater we took various rimes and looked at the a s that were quadratic residues and the a s that were nonresidues. For examle, we made a table of squares
More informationMath 124 Homework 5 Solutions
Math 12 Homework 5 Solutions by Luke Gustafson Fall 2003 1. 163 1 2 (mod 2 gives = 2 the smallest rime. 2a. First, consider = 2. We know 2 is not a uadratic residue if and only if 3, 5 (mod 8. By Dirichlet
More informationSection 16.3 Double Integrals over General Regions
Section 6.3 Double Integrls over Generl egions Not ever region is rectngle In the lst two sections we considered the problem of integrting function of two vribles over rectngle. This sitution however is
More informationMT 430 Intro to Number Theory MIDTERM 2 PRACTICE
MT 40 Intro to Number Theory MIDTERM 2 PRACTICE Material covered Midterm 2 is comrehensive but will focus on the material of all the lectures from February 9 u to Aril 4 Please review the following toics
More informationQuadratic Residues. Legendre symbols provide a computational tool for determining whether a quadratic congruence has a solution. = a (p 1)/2 (mod p).
Quadratic Residues 4015 a is a quadratic residue mod m if x = a (mod m). Otherwise, a is a quadratic nonresidue. Quadratic Recirocity relates the solvability of the congruence x = (mod q) to the solvability
More informationMTH 3527 Number Theory Quiz 10 (Some problems that might be on the quiz and some solutions.) 1. Euler φfunction. Desribe all integers n such that:
MTH 7 Number Theory Quiz 10 (Some roblems that might be on the quiz and some solutions.) 1. Euler φfunction. Desribe all integers n such that: (a) φ(n) = Solution: n = 4,, 6 since φ( ) = ( 1) =, φ() =
More informationExample: Modulo 11: Since Z p is cyclic, there is a generator. Let g be a generator of Z p.
Qudrtic Residues Defiitio: The umbers 0, 1,,, ( mod, re clled udrtic residues modulo Numbers which re ot udrtic residues modulo re clled udrtic oresidues modulo Exmle: Modulo 11: Itroductio to Number
More informationIntroduction to Number Theory 2. c Eli Biham  November 5, Introduction to Number Theory 2 (12)
Introduction to Number Theory c Eli Biham  November 5, 006 345 Introduction to Number Theory (1) Quadratic Residues Definition: The numbers 0, 1,,...,(n 1) mod n, are called uadratic residues modulo n.
More informationDomination and Independence on Square Chessboard
Engineering nd Technology Journl Vol. 5, Prt, No. 1, 017 A.A. Omrn Deprtment of Mthemtics, College of Eduction for Pure Science, University of bylon, bylon, Irq pure.hmed.omrn@uobby lon.edu.iq Domintion
More informationSome Connections Between Primitive Roots and Quadratic NonResidues Modulo a Prime
Some Connections Between Primitive Roots nd Qudrtic NonResidues Modulo Prime Sorin Iftene Dertment of Comuter Science Al. I. Cuz University Isi, Romni Emil: siftene@info.uic.ro Abstrct In this er we resent
More informationSpiral Tilings with Ccurves
Spirl Tilings with curves Using ombintorics to Augment Trdition hris K. Plmer 19 North Albny Avenue hicgo, Illinois, 0 chris@shdowfolds.com www.shdowfolds.com Abstrct Spirl tilings used by rtisns through
More informationTo be able to determine the quadratic character of an arbitrary number mod p (p an odd prime), we. The first (and most delicate) case concerns 2
Quadratic Reciprocity To be able to determine the quadratic character of an arbitrary number mod p (p an odd prime), we need to be able to evaluate q for any prime q. The first (and most delicate) case
More informationPolar Coordinates. July 30, 2014
Polr Coordintes July 3, 4 Sometimes it is more helpful to look t point in the xyplne not in terms of how fr it is horizontlly nd verticlly (this would men looking t the Crtesin, or rectngulr, coordintes
More informationCS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates
Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: www.ses.gwu.edu/~bhgiweb/cs35/ Digitl Logic Circuits We sw how we cn build the
More information(CATALYST GROUP) B"sic Electric"l Engineering
(CATALYST GROUP) B"sic Electric"l Engineering 1. Kirchhoff s current l"w st"tes th"t (") net current flow "t the junction is positive (b) Hebr"ic sum of the currents meeting "t the junction is zero (c)
More informationExample. Check that the Jacobian of the transformation to spherical coordinates is
lss, given on Feb 3, 2, for Mth 3, Winter 2 Recll tht the fctor which ppers in chnge of vrible formul when integrting is the Jcobin, which is the determinnt of mtrix of first order prtil derivtives. Exmple.
More informationOn the Fibonacci Sequence. By: Syrous Marivani LSUA. Mathematics Department. Alexandria, LA 71302
On the Fibonacci Sequence By: Syrous Marivani LSUA Mathematics Deartment Alexandria, LA 70 The socalled Fibonacci sequence {(n)} n 0 given by: (n) = (n ) + (n ), () where (0) = 0, and () =. The ollowing
More informationFrancis Gaspalou Second edition of February 10, 2012 (First edition on January 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY?
Frncis Gslou Second edition of Ferury 10, 2012 (First edition on Jnury 28, 2012) HOW MANY SQUARES ARE THERE, Mr TARRY? ABSTRACT In this er, I enumerte ll the 8x8 imgic sures given y the Trry s ttern. This
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem Theorem. Let n 1,..., n r be r positive integers relatively prime in pairs. (That is, gcd(n i, n j ) = 1 whenever 1 i < j r.) Let a 1,..., a r be any r integers. Then the
More informationNUMBER THEORY AMIN WITNO
NUMBER THEORY AMIN WITNO.. w w w. w i t n o. c o m Number Theory Outlines and Problem Sets Amin Witno Preface These notes are mere outlines for the course Math 313 given at Philadelphia
More informationLecture 20. Intro to line integrals. Dan Nichols MATH 233, Spring 2018 University of Massachusetts.
Lecture 2 Intro to line integrls Dn Nichols nichols@mth.umss.edu MATH 233, Spring 218 University of Msschusetts April 12, 218 (2) onservtive vector fields We wnt to determine if F P (x, y), Q(x, y) is
More informationSection 17.2: Line Integrals. 1 Objectives. 2 Assignments. 3 Maple Commands. 1. Compute line integrals in IR 2 and IR Read Section 17.
Section 7.: Line Integrls Objectives. ompute line integrls in IR nd IR 3. Assignments. Red Section 7.. Problems:,5,9,,3,7,,4 3. hllenge: 6,3,37 4. Red Section 7.3 3 Mple ommnds Mple cn ctully evlute line
More information6. Find an inverse of a modulo m for each of these pairs of relatively prime integers using the method
Exercises Exercises 1. Show that 15 is an inverse of 7 modulo 26. 2. Show that 937 is an inverse of 13 modulo 2436. 3. By inspection (as discussed prior to Example 1), find an inverse of 4 modulo 9. 4.
More informationVector Calculus. 1 Line Integrals
Vector lculus 1 Line Integrls Mss problem. Find the mss M of very thin wire whose liner density function (the mss per unit length) is known. We model the wire by smooth curve between two points P nd Q
More informationMETHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN. Inventor: Brian L. Baskin
METHOD OF LOCATION USING SIGNALS OF UNKNOWN ORIGIN Inventor: Brin L. Bskin 1 ABSTRACT The present invention encompsses method of loction comprising: using plurlity of signl trnsceivers to receive one or
More information10.4 AREAS AND LENGTHS IN POLAR COORDINATES
65 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES.4 AREAS AND LENGTHS IN PLAR CRDINATES In this section we develop the formul for the re of region whose oundry is given y polr eqution. We need to use the
More informationThe Chinese Remainder Theorem
The Chinese Remainder Theorem Theorem. Let m and n be two relatively prime positive integers. Let a and b be any two integers. Then the two congruences x a (mod m) x b (mod n) have common solutions. Any
More information(1) Primary Trigonometric Ratios (SOH CAH TOA): Given a right triangle OPQ with acute angle, we have the following trig ratios: ADJ
Tringles nd Trigonometry Prepred y: S diyy Hendrikson Nme: Dte: Suppose we were sked to solve the following tringles: Notie tht eh tringle hs missing informtion, whih inludes side lengths nd ngles. When
More informationEE Controls Lab #2: Implementing StateTransition Logic on a PLC
Objective: EE 44  Controls Lb #2: Implementing Stternsition Logic on PLC ssuming tht speed is not of essence, PLC's cn be used to implement stte trnsition logic. he dvntge of using PLC over using hrdwre
More informationFirst Round Solutions Grades 4, 5, and 6
First Round Solutions Grdes 4, 5, nd 1) There re four bsic rectngles not mde up of smller ones There re three more rectngles mde up of two smller ones ech, two rectngles mde up of three smller ones ech,
More informationPolar coordinates 5C. 1 a. a 4. π = 0 (0) is a circle centre, 0. and radius. The area of the semicircle is π =. π a
Polr coordintes 5C r cos Are cos d (cos + ) sin + () + 8 cos cos r cos is circle centre, nd rdius. The re of the semicircle is. 8 Person Eduction Ltd 8. Copying permitted for purchsing institution only.
More informationLECTURE 3: CONGRUENCES. 1. Basic properties of congruences We begin by introducing some definitions and elementary properties.
LECTURE 3: CONGRUENCES 1. Basic properties of congruences We begin by introducing some definitions and elementary properties. Definition 1.1. Suppose that a, b Z and m N. We say that a is congruent to
More informationChapter 5 Analytic Trigonometry
Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos / sec, sec / cos, nd tn sin / cos. sin / csc nd tn / cot 3. csc / sin, cot / tn, nd cot cos
More informationTriangles and parallelograms of equal area in an ellipse
1 Tringles nd prllelogrms of equl re in n ellipse Roert Buonpstore nd Thoms J Osler Mthemtics Deprtment RownUniversity Glssoro, NJ 0808 USA uonp0@studentsrownedu osler@rownedu Introduction In the pper
More informationMAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES
MAXIMUM FLOWS IN FUZZY NETWORKS WITH FUNNELSHAPED NODES Romn V. Tyshchuk Informtion Systems Deprtment, AMI corportion, Donetsk, Ukrine Emil: rt_science@hotmil.com 1 INTRODUCTION During the considertion
More informationChapter 5 Analytic Trigonometry
Section 5. Fundmentl Identities 03 Chter 5 Anlytic Trigonometry Section 5. Fundmentl Identities Exlortion. cos > sec, sec > cos, nd tn sin > cos. sin > csc nd tn > cot 3. csc > sin, cot > tn, nd cot cos
More informationCHAPTER 2 LITERATURE STUDY
CHAPTER LITERATURE STUDY. Introduction Multipliction involves two bsic opertions: the genertion of the prtil products nd their ccumultion. Therefore, there re two possible wys to speed up the multipliction:
More informationGeometric quantities for polar curves
Roerto s Notes on Integrl Clculus Chpter 5: Bsic pplictions of integrtion Section 10 Geometric quntities for polr curves Wht you need to know lredy: How to use integrls to compute res nd lengths of regions
More informationCongruences for Stirling Numbers of the Second Kind Modulo 5
Southest Asin Bulletin of Mthemtics (2013 37: 795 800 Southest Asin Bulletin of Mthemtics c SEAMS. 2013 Congruences for Stirling Numbers of the Second Kind Modulo 5 Jinrong Zho School of Economic Mthemtics,
More informationMath Circles Finite Automata Question Sheet 3 (Solutions)
Mth Circles Finite Automt Question Sheet 3 (Solutions) Nickols Rollick nrollick@uwterloo.c Novemer 2, 28 Note: These solutions my give you the nswers to ll the prolems, ut they usully won t tell you how
More informationUNIVERSITY OF MANITOBA DATE: December 7, FINAL EXAMINATION TITLE PAGE TIME: 3 hours EXAMINER: M. Davidson
TITLE PAGE FAMILY NAME: (Print in ink) GIVEN NAME(S): (Print in ink) STUDENT NUMBER: SEAT NUMBER: SIGNATURE: (in ink) (I understand that cheating is a serious offense) INSTRUCTIONS TO STUDENTS: This is
More information9.4. ; 65. A family of curves has polar equations. ; 66. The astronomer Giovanni Cassini ( ) studied the family of curves with polar equations
54 CHAPTER 9 PARAMETRIC EQUATINS AND PLAR CRDINATES 49. r, 5. r sin 3, 5 54 Find the points on the given curve where the tngent line is horizontl or verticl. 5. r 3 cos 5. r e 53. r cos 54. r sin 55. Show
More informationConjectures and Results on Super Congruences
Conjectures and Results on Suer Congruences ZhiWei Sun Nanjing University Nanjing 210093, P. R. China zwsun@nju.edu.cn htt://math.nju.edu.cn/ zwsun Feb. 8, 2010 Part A. Previous Wor by Others What are
More informationSOLVING TRIANGLES USING THE SINE AND COSINE RULES
Mthemtics Revision Guides  Solving Generl Tringles  Sine nd Cosine Rules Pge 1 of 17 M.K. HOME TUITION Mthemtics Revision Guides Level: GCSE Higher Tier SOLVING TRIANGLES USING THE SINE AND COSINE RULES
More informationExercise 11. The Sine Wave EXERCISE OBJECTIVE DISCUSSION OUTLINE. Relationship between a rotating phasor and a sine wave DISCUSSION
Exercise 11 The Sine Wve EXERCISE OBJECTIVE When you hve completed this exercise, you will be fmilir with the notion of sine wve nd how it cn be expressed s phsor rotting round the center of circle. You
More informationFermat s little theorem. RSA.
.. Computing large numbers modulo n (a) In modulo arithmetic, you can always reduce a large number to its remainder a a rem n (mod n). (b) Addition, subtraction, and multiplication preserve congruence:
More informationTheme: Don t get mad. Learn mod.
FERURY When 1 is divided by 5, the reminder is. nother wy to sy this is opyright 015 The Ntionl ouncil of Techers of Mthemtics, Inc. www.nctm.org. ll rights reserved. This mteril my not be copied or distributed
More informationUnit 1: Chapter 4 Roots & Powers
Unit 1: Chpter 4 Roots & Powers Big Ides Any number tht cn be written s the frction mm, nn 0, where m nd n re integers, is nn rtionl. Eponents cn be used to represent roots nd reciprocls of rtionl numbers.
More informationSOLUTIONS TO PROBLEM SET 5. Section 9.1
SOLUTIONS TO PROBLEM SET 5 Section 9.1 Exercise 2. Recall that for (a, m) = 1 we have ord m a divides φ(m). a) We have φ(11) = 10 thus ord 11 3 {1, 2, 5, 10}. We check 3 1 3 (mod 11), 3 2 9 (mod 11), 3
More informationFP2 POLAR COORDINATES: PAST QUESTIONS
FP POLAR COORDINATES: PAST QUESTIONS. The curve C hs polr eqution r = cosθ, () Sketch the curve C. () (b) Find the polr coordintes of the points where tngents to C re prllel to the initil line. (6) (c)
More informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationCollection of rules, techniques and theorems for solving polynomial congruences 11 April 2012 at 22:02
Collection of rules, techniques and theorems for solving polynomial congruences 11 April 2012 at 22:02 Public Polynomial congruences come up constantly, even when one is dealing with much deeper problems
More informationNumber Theory  Divisibility Number Theory  Congruences. Number Theory. June 23, Number Theory
 Divisibility  Congruences June 23, 2014 Primes  Divisibility  Congruences Definition A positive integer p is prime if p 2 and its only positive factors are itself and 1. Otherwise, if p 2, then p
More informationKirchhoff s Rules. Kirchhoff s Laws. Kirchhoff s Rules. Kirchhoff s Laws. Practice. Understanding SPH4UW. Kirchhoff s Voltage Rule (KVR):
SPH4UW Kirchhoff s ules Kirchhoff s oltge ule (K): Sum of voltge drops round loop is zero. Kirchhoff s Lws Kirchhoff s Current ule (KC): Current going in equls current coming out. Kirchhoff s ules etween
More informationUnderstanding Basic Analog Ideal Op Amps
Appliction Report SLAA068A  April 2000 Understnding Bsic Anlog Idel Op Amps Ron Mncini Mixed Signl Products ABSTRACT This ppliction report develops the equtions for the idel opertionl mplifier (op mp).
More informationSeven Sisters. Visit for video tutorials
Seven Sisters This imge is from www.quiltstudy.org. Plese visit this website for more informtion on Seven Sisters quilt ptterns. Visit www.blocloc.com for video tutorils 1 The Seven Sisters design cn be
More informationMesh and Node Equations: More Circuits Containing Dependent Sources
Mesh nd Node Equtions: More Circuits Contining Dependent Sources Introduction The circuits in this set of problems ech contin single dependent source. These circuits cn be nlyzed using mesh eqution or
More information13.1 Double Integral over Rectangle. f(x ij,y ij ) i j I <ɛ. f(x, y)da.
CHAPTE 3, MULTIPLE INTEGALS Definition. 3. Double Integrl over ectngle A function f(x, y) is integrble on rectngle [, b] [c, d] if there is number I such tht for ny given ɛ>0thereisδ>0 such tht, fir ny
More informationLECTURE 7: POLYNOMIAL CONGRUENCES TO PRIME POWER MODULI
LECTURE 7: POLYNOMIAL CONGRUENCES TO PRIME POWER MODULI 1. Hensel Lemma for nonsingular solutions Although there is no analogue of Lagrange s Theorem for prime power moduli, there is an algorithm for determining
More informationFoundations of Cryptography
Foundations of Cryptography Ville Junnila viljun@utu.fi Department of Mathematics and Statistics University of Turku 2015 Ville Junnila viljun@utu.fi Lecture 10 1 of 17 The order of a number (mod n) Definition
More informationb) Find all positive integers smaller than 200 which leave remainder 1, 3, 4 upon division by 3, 5, 7 respectively.
Solutions to Exam 1 Problem 1. a) State Fermat s Little Theorem and Euler s Theorem. b) Let m, n be relatively prime positive integers. Prove that m φ(n) + n φ(m) 1 (mod mn). Solution: a) Fermat s Little
More informationSection 10.2 Graphing Polar Equations
Section 10.2 Grphing Polr Equtions OBJECTIVE 1: Sketching Equtions of the Form rcos, rsin, r cos r sin c nd Grphs of Polr Equtions of the Form rcos, rsin, r cos r sin c, nd where,, nd c re constnts. The
More informationECE 274 Digital Logic. Digital Design. Datapath Components Shifters, Comparators, Counters, Multipliers Digital Design
ECE 27 Digitl Logic Shifters, Comprtors, Counters, Multipliers Digitl Design..7 Digitl Design Chpter : Slides to ccompny the textbook Digitl Design, First Edition, by Frnk Vhid, John Wiley nd Sons Publishers,
More information30 HWASIN PARK, JOONGSOO PARK AND DAEYEOUL KIM Lemma 1.1. Let =2 k q +1, k 2 Z +. Then the set of rimitive roots modulo is the set of quadratic nonre
J. KSIAM Vol.4, No.1, 2938, 2000 A CRITERION ON PRIMITIVE ROOTS MODULO Hwasin Park, Joongsoo Park and Daeyeoul Kim Abstract. In this aer, we consider a criterion on rimitive roots modulo where is the
More informationNONCLASSICAL CONSTRUCTIONS II
NONLSSIL ONSTRUTIONS II hristopher Ohrt UL Mthcircle  Nov. 22, 2015 Now we will try ourselves on onceletsteiner constructions. You cn only use n (unmrked) strightedge but you cn ssume tht somewhere
More informationINTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS
CHAPTER 8 INTRODUCTION TO TRIGONOMETRY AND ITS APPLICATIONS (A) Min Concepts nd Results Trigonometric Rtios of the ngle A in tringle ABC right ngled t B re defined s: sine of A = sin A = side opposite
More informationStudent Book SERIES. Patterns and Algebra. Name
E Student Book 3 + 7 5 + 5 Nme Contents Series E Topic Ptterns nd functions (pp. ) identifying nd creting ptterns skip counting completing nd descriing ptterns predicting repeting ptterns predicting growing
More informationAQA Level 2 Further mathematics Further algebra. Section 3: Inequalities and indices
AQA Level Further mthemtics Further lgebr Sectio : Iequlities d idices Notes d Emples These otes coti subsectios o Iequlities Lier iequlities Qudrtic iequlities Multiplyig epressios The rules of idices
More informationPractice Midterm 2 Solutions
Practice Midterm 2 Solutions May 30, 2013 (1) We want to show that for any odd integer a coprime to 7, a 3 is congruent to 1 or 1 mod 7. In fact, we don t need the assumption that a is odd. By Fermat s
More informationSOLUTIONS FOR PROBLEM SET 4
SOLUTIONS FOR PROBLEM SET 4 A. A certain integer a gives a remainder of 1 when divided by 2. What can you say about the remainder that a gives when divided by 8? SOLUTION. Let r be the remainder that a
More informationSolutions for the Practice Questions
Solutions for the Practice Questions Question 1. Find all solutions to the congruence 13x 12 (mod 35). Also, answer the following questions about the solutions to the above congruence. Are there solutions
More informationAn interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g.,
Binary exponentiation An interesting class of problems of a computational nature ask for the standard residue of a power of a number, e.g., What are the last two digits of the number 2 284? In the absence
More informationPrimitive Roots. Chapter Orders and Primitive Roots
Chapter 5 Primitive Roots The name primitive root applies to a number a whose powers can be used to represent a reduced residue system modulo n. Primitive roots are therefore generators in that sense,
More informationGeneral Augmented Rook Boards & Product Formulas
Forml Power Series nd Algebric Combintorics Séries Formelles et Combintoire Algébriue Sn Diego, Cliforni 006 Generl Augmented Rook Bords & Product Formuls Brin K Miceli Abstrct There re number of soclled
More informationALGEBRA: Chapter I: QUESTION BANK
1 ALGEBRA: Chapter I: QUESTION BANK Elements of Number Theory Congruence One mark questions: 1 Define divisibility 2 If a b then prove that a kb k Z 3 If a b b c then PT a/c 4 If a b are two non zero integers
More informationDetermine currents I 1 to I 3 in the circuit of Fig. P2.14. Solution: For the loop containing the 18V source, I 1 = 0.
Prolem.14 Determine currents 1 to 3 in the circuit of Fig. P.14. 1 A 18 V Ω 3 A 1 8 Ω 1 Ω 7 Ω 4 Ω 3 Figure P.14: Circuit for Prolem.14. For the loop contining the 18V source, Hence, 1 = 1.5 A. KCL t node
More informationAlgebra Practice. Dr. Barbara Sandall, Ed.D., and Travis Olson, M.S.
By Dr. Brr Sndll, Ed.D., Dr. Melfried Olson, Ed.D., nd Trvis Olson, M.S. COPYRIGHT 2006 Mrk Twin Medi, Inc. ISBN 9781580377546 Printing No. 404042EB Mrk Twin Medi, Inc., Pulishers Distriuted y CrsonDellos
More informationDiscrete Square Root. Çetin Kaya Koç Winter / 11
Discrete Square Root Çetin Kaya Koç koc@cs.ucsb.edu Çetin Kaya Koç http://koclab.cs.ucsb.edu Winter 2017 1 / 11 Discrete Square Root Problem The discrete square root problem is defined as the computation
More informationModule 9. DC Machines. Version 2 EE IIT, Kharagpur
Module 9 DC Mchines Version EE IIT, Khrgpur esson 40 osses, Efficiency nd Testing of D.C. Mchines Version EE IIT, Khrgpur Contents 40 osses, efficiency nd testing of D.C. mchines (esson40) 4 40.1 Gols
More informationMisty. Sudnow Dot Songs
Sudnow Dot Songs isty T The Dot Song is nottionl system tht depicts voiced chords in wy where the nonmusic reder cn find these firly redily. But the Dot Song is not intended be red, not s sight reder
More informationHomework #1 due Monday at 6pm. White drop box in Student Lounge on the second floor of Cory. Tuesday labs cancelled next week
Announcements Homework #1 due Mondy t 6pm White drop ox in Student Lounge on the second floor of Cory Tuesdy ls cncelled next week Attend your other l slot Books on reserve in Bechtel Hmley, 2 nd nd 3
More informationb = and their properties: b 1 b 2 b 3 a b is perpendicular to both a and 1 b = x = x 0 + at y = y 0 + bt z = z 0 + ct ; y = y 0 )
***************** Disclimer ***************** This represents very brief outline of most of the topics covered MA261 *************************************************** I. Vectors, Lines nd Plnes 1. Vector
More informationMath 412: Number Theory Lecture 6: congruence system and
Math 412: Number Theory Lecture 6: congruence system and classes Gexin Yu gyu@wm.edu College of William and Mary Chinese Remainder Theorem Chinese Remainder Theorem: let m 1, m 2,..., m k be pairwise coprimes.
More information& Y Connected resistors, Light emitting diode.
& Y Connected resistors, Light emitting diode. Experiment # 02 Ojectives: To get some hndson experience with the physicl instruments. To investigte the equivlent resistors, nd Y connected resistors, nd
More informationWI1402LR Calculus II Delft University of Technology
WI402LR lculus II elft University of Technology Yer 203 204 Michele Fcchinelli Version.0 Lst modified on Februry, 207 Prefce This summry ws written for the course WI402LR lculus II, tught t the elft
More informationEnergy Harvesting TwoWay Channels With Decoding and Processing Costs
IEEE TRANSACTIONS ON GREEN COMMUNICATIONS AND NETWORKING, VOL., NO., MARCH 07 3 Energy Hrvesting TwoWy Chnnels With Decoding nd Processing Costs Ahmed Arf, Student Member, IEEE, Abdulrhmn Bknin, Student
More informationAbacabaDabacaba! by Michael Naylor Western Washington University
AbcbDbcb! by Michel Nylor Western Wshington University The Abcb structure shows up in n mzing vriety of plces. This rticle explores 10 surprising ides which ll shre this pttern, pth tht will tke us through
More informationOutline Introduction Big Problems that Brun s Sieve Attacks Conclusions. Brun s Sieve. Joe Fields. November 8, 2007
Big Problems that Attacks November 8, 2007 Big Problems that Attacks The Sieve of Eratosthenes The Chinese Remainder Theorem picture Big Problems that Attacks Big Problems that Attacks Eratosthene s Sieve
More informationTopic 20: Huffman Coding
Topic 0: Huffmn Coding The uthor should gze t Noh, nd... lern, s they did in the Ark, to crowd gret del of mtter into very smll compss. Sydney Smith, dinburgh Review Agend ncoding Compression Huffmn Coding
More informationMONOCHRONICLE STRAIGHT
UPDATED 092010 HYDROCARBON Hydrocrbon is ponchostyle cowl in bulkyweight yrn, worked in the round. It ws designed to be s prcticl s it is stylish, with shping tht covers the neck nd shoulders nd the
More informationThe congruence relation has many similarities to equality. The following theorem says that congruence, like equality, is an equivalence relation.
Congruences A congruence is a statement about divisibility. It is a notation that simplifies reasoning about divisibility. It suggests proofs by its analogy to equations. Congruences are familiar to us
More informationWilson s Theorem and Fermat s Theorem
Wilson s Theorem and Fermat s Theorem 7272006 Wilson s theorem says that p is prime if and only if (p 1)! = 1 (mod p). Fermat s theorem says that if p is prime and p a, then a p 1 = 1 (mod p). Wilson
More informationMath 127: Equivalence Relations
Math 127: Equivalence Relations Mary Radcliffe 1 Equivalence Relations Relations can take many forms in mathematics. In these notes, we focus especially on equivalence relations, but there are many other
More informationSynchronous Generator Line Synchronization
Synchronous Genertor Line Synchroniztion 1 Synchronous Genertor Line Synchroniztion Introduction One issue in power genertion is synchronous genertor strting. Typiclly, synchronous genertor is connected
More informationJoanna Towler, Roading Engineer, Professional Services, NZTA National Office Dave Bates, Operations Manager, NZTA National Office
. TECHNICA MEMOANDM To Cc repred By Endorsed By NZTA Network Mngement Consultnts nd Contrctors NZTA egionl Opertions Mngers nd Are Mngers Dve Btes, Opertions Mnger, NZTA Ntionl Office Jonn Towler, oding
More information