Quadratic reciprocity
|
|
- Erik Patrick
- 6 years ago
- Views:
Transcription
1 Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition 1. Modulo Nottion Let, e integer numers nd n ositive integer numer, then we sy tht is congruent to modulo n nd we write mod if n divides the difference. Also recll the Fundmentl theorem of rithmetic: Every integer n cn e written uniquely s n ± r r where r is ositive integer greter thn or equl to 1, 1,... r distinct rime numers nd 1,... r re ositive integers. The lst nd non-trivil result tht we hve to recll in order to roceed further is Fermt s Little Theorem which sys tht if is n integer nd is rime, then mod, or equivlently 1 1 mod if, 1. Now we cn strt tlking out the qudrtic recirocity method. This method ws develoed intuitively not ctully on this modern form tht you will see here y some gret numer theorists of the 17th nd 18th century like Fermt, Euler, Lgrnge nd Legendre. They egn y looking t the qudrtic olynomils modulo rime nd trying to solve it in the most generl forms. Nmely, they did not try to solve x + x + c 0 in the rel or comlex numers, which ws lredy known, ut rther to solve the eqution x + x + c 0mod where is rime numer. in the integer numers. We will cll such n eqution solvle, if there exists n integer x 0 such tht x 0 + x 0 + c 0 mod. Although with little success, these mthemticins stted some imortnt conjectures in this field tht would rodened the field of numer theory. A mjor rekthrough in this direction cme when Guss in 1798 roved wht 1
2 is now clled the Qudrtic Recirocity Lw, nmely, if, q re rime numers nd if q 1 mod 4, then x 0 mod q is solvle if nd only if x q 0 mod is solvle nd if q 1 mod 3, then x 0 mod q is solvle if nd only if x + q 0 mod is solvle. We will lern this theorem lter, ut in more modern formultion. Now let s discuss out the essentil results of this toic. Definition. Let m, n nd e integers, m 1, n 1 nd, m 1. We sy tht is residue of n-th degree modulo m if congruence x n mod m hs n integer solution; else is nonresidue of n-th degree. In rticulr, if n, we will cll qudrtic residue or qudrtic nonresidue. Now we stte our first theorem: Theorem 3. Given rime nd n integer, the eqution x hs zero, one, or two solutions modulo. Proof. Suose tht the considered congruence hs solution x 1. Then so clerly is x x 1. There re no other solutions modulo, ecuse x x 1 mod imlies x ±x 1. As n immedite corollry of the ove thereom, we hve tht for every odd rime, mong the numers 1,,..., 1 there re exctly 1 qudrtic residues nd s mny qudrtic nonresidues. Now we re redy to define the Legendre s symol: Definition 4. Given rime numer nd n integer, Legendres symol is defined s 1, if nd is qudrtic residue mod ; 1, if nd is qudrtic nonresidue mod ; 0, if. x As first result out the Legendre s symol, we see tht 1 for ech rime nd integer x, x. From now on, unless noted otherwise, is lwys n odd rime nd n integer. Clerly, is qudrtic residue modulo if nd only if so is + k for some integer k. Thus we my regrd Legendre s symol s function from the residue clsses modulo to the set { 1, 0, 1}. Fermts theorem sserts tht if, 1 then 1 1 mod, which imlies 1 ±1 mod. More recisely: Theorem 5. Euler s Criterion If is n integer nd rime, then 1 mod.
3 3 Proof. The sttement is trivil for divides. From now on we ssume tht. Let g e rimitive root modulo. Then the numers g i, i 0, 1,..., form reduced system of residues modulo. We oserve tht g i 1 g i 1 1mod if nd only if 1 divides i 1, or equivlently, divides i. On the other hnd, g i is qudrtic residue modulo if nd only if there exists j {0, 1,, } such tht g j g i mod, which is equivlent to j i mod 1. The lst congruence is solvle if nd only if divides i, tht is, exctly when g i 1 1 mod. Now, y Euler s Criterion we hve tht 1 1 1, for ll integers, nd rime numers, therefore we roved tht the Legendre s symol is multilictive. Theorem 6. for ll integers, nd rime numer. From Euler s Criterion, we lso get wht is clled the Theorem 7. First Sulement of the Qudrtic Recirocity Lw For every rime numer 3, Now we return to our initil rolem, the one tht strted ll this theory out qudrtic residues: Theorem 8. Let x, y e corime integers nd,, c e ritrry integers. If is n odd rime divisor of numer x +xy +cy which doesn t divide c, then D 4c is qudrtic residue modulo. In rticulr, if divides x Dy nd x, y 1, then D is qudrtic residue modulo. Proof. Denote N x + xy + cy. Since 4N x + y Dy, we hve x + y Dy mod. Furthermore, y is not divisile y ; otherwise so would e x+y nd therefore x itself, contrdicting the ssumtion. There is n integer y 1 such tht yy 1 1mod. Multilying the ove congruence y y 1 gives us xy 1 + yy 1 Dyy 1 Dmod, imlying the sttement. Let us get closer to the min theorem of this introduction to qudrtic recirocity, ut first we hve to stte nother intermedite theorem: Theorem 9. Second Sulement of the Qudrtic Recirocity Lw We hve In other words, is qudrtic residue modulo rime 3 if nd only if ±1 mod 8.
4 4 The roof is retty comlicted nd uses the Guss Lemm, very eutiful nd helful trick. With ll this eing sid, we conclude with the most imortnt theorem of this rt, the Guss Lw of Qudrtic Recirocity: Theorem 10. For ny different odd rimes nd q, q 1 1 q 1. q The roof of this fct is lso retty tricky, nd it will e ommitted. Nevertheless, this won t sto us to use it frequently in deducing other roerties out Legendre s symol. You my sk why Guss did not stte his theorem in this more elegnt form. Well, you should know tht Legendre invented his symol long fter Guss roved the qudrtic recirocity lw, similrly s Guss invented the modulo sign long fter Fermt, Euler nd Lgrnge roved their theorems. Exercises 1. Wht is 1 if is rime numer of the form 19k + 3? Is 5 qudrtic residue modulo 79? Is 6 qudrtic residue modulo 79? Is qudrtic trust me, this numer is indeed rime?. Comute the following Legendre symols: In fct, wht Guss roved ws 1 1 8, , , 1 [ +1 4 ]. Prove tht [ ] + 1 mod 4 for ny odd rime numer, so indeed Theorem 9 is exctly wht Guss roved. Note tht [x] is the gretest integer smller or equl to x 4. Is 3 squre mod 41? Is 15 squre mod 41? CHMMC, Winter Comute the numer of rimes less thn 100 such tht divides n +n+1 for some integer n. CHMMC, Winter Find ll the rimes with the roerty tht is erfect squre. Junior Blkn MO 007Hint: Use Fermt s Little Theorem 7.
5 5 is qudrtic residue modulo if nd only if 1 or 3 mod 8; 3 is qudrtic residue modulo if nd only if 1 mod 6; c 3 is qudrtic residue modulo if nd only if ±1 mod 1; d 5 is qudrtic residue modulo if nd only if ±1 mod Show tht there exist infinitely mny rime numers of the form 10k Prove tht for n N every rime divisor of numer n 4 n + 1 is of the form 1k If is rime of the form 4k + 1, rove tht x 1! is solution of the congruence x mod. Here m! 1... m 11. There exists nturl numer < +1 tht is qudrtic nonresidue modulo [. Hint: Suose is qudrtic nonresidue. Wht cn you sy out ] + 1? 1. Chllenge rolem Prove tht n integer is qudrtic residue modulo every rime numer if nd only if is erfect squre. 13. Evlute [ ] [ ] [ ] [ Note tht [x] is the gretest integer smller or equl to x 14. Prove Guss Lemm: Let e rime numer nd n integer tht is corime with. Consider the integers,, 3,..., 1 nd tke their lest ositive residue modulo. These residues re ll distinct, so there re 1 of them. Let m e the numer of qudrtic residues tht re greter thn. Show tht 1 m. Generliztion: The Jcoi Symol Definition 11. Let e n integer nd n odd numer, nd let α1 1 α1... αr r e the fctoriztion of onto rimes. Jcois symol is defined s roduct of Legendre s symols, nmely α1 1 α αr. r We see tht if is qudrtic residue modulo n, then clerly n 1. However the converse is not true nd we cn find counterexmle. For exmle, , ].
6 6 ut is not qudrtic residue modulo 15, s is not so modulo 3 nd 5. Nevertheless, if is qudrtic nonresidue modulo n, then we get n 1, which imlies tht there exists rime numer dividing n such tht 1 y the ove definition. All this discussion cn e summrrized in the following sttement: Theorem 1. Let e n integer nd ositive integer, nd let α1 1 α... αr r e the fctoriztion of onto rimes. Then is qudrtic residue modulo if nd only if is qudrtic residue modulo αi i for ech i 1,,..., r. We will not see the roof of this theorem here. One direction is trivil, nmely if there exists n x such tht x mod n, then clerly the sme x stisfies x mod αi i. The other direction should e n esy exercise for those of you who know the Chinese Reminder Theorem. The most eutiful thing out the Jcoi symol is tht it oeys most of the lws tht the Legendre s Symol is oeying. Theorem 13. For ll integers, nd odd numers c, d the following equlities hold: + c c c 1 c c c cd c d 3 We lso hve tht the Jcoi symol oeys the three reciroicity lws, nmely Theorem 14. For every odd integer, 1 1 1, nd for ny two corime odd numers, it holds tht Exercises 1. If gcd, nm1, then nm n.. Let F x x 17x 19x 33. Prove tht for ech m ositive integer, the eqution F x 0 mod m
7 7 hs solution x in N. But F x 0 doesn t hve ny integer solutions not even rtionl solutions. 3. Let m, n 3 e ositive odd integers. Prove tht m 1 doesn t divide 3 n Show tht there re no ositive integers x, y, z, t such tht x + y + t 4xyz. Hint: Write the eqution s 4zt + 1 4zy 14zx 1; now look t the Jcoi symol. z 4yz 1 5. Chllenge Prolem The numer of qudrtic residues modulo n n 1 is equl to [ n 1 ] [ 1 n+1 ] 1 + for, nd + 1 for Chllenge Prolem Prove tht the eqution x y 3 5 hs no integer solutions x, y. 7. Chllenge Prolem Prove tht 4kxy 1 does not divide the numer x m + y n for ny ositive integers x, y, k, m, n.
QUADRATIC RESIDUES MATH 372. FALL INSTRUCTOR: PROFESSOR AITKEN
QUADRATIC RESIDUES MATH 37 FALL 005 INSTRUCTOR: PROFESSOR AITKEN When is n integer sure modulo? When does udrtic eution hve roots modulo? These re the uestions tht will concern us in this hndout 1 The
More informationQuadratic Residues. Chapter Quadratic residues
Chter 8 Qudrtic Residues 8. Qudrtic residues Let n>be given ositive integer, nd gcd, n. We sy tht Z n is qudrtic residue mod n if the congruence x mod n is solvble. Otherwise, is clled qudrtic nonresidue
More informationDuke Math Meet
Duke Mth Meet 01-14 Power Round Qudrtic Residues nd Prime Numers For integers nd, we write to indicte tht evenly divides, nd to indicte tht does not divide For exmle, 4 nd 4 Let e rime numer An integer
More informationSupplement 4 Permutations, Legendre symbol and quadratic reciprocity
Sulement 4 Permuttions, Legendre symbol nd qudrtic recirocity 1. Permuttions. If S is nite set contining n elements then ermuttion of S is one to one ming of S onto S. Usully S is the set f1; ; :::; ng
More informationKronecker-Jacobi symbol and Quadratic Reciprocity. Q b /Q p
Kronecker-Jcoi symol nd Qudrtic Recirocity Let Q e the field of rtionl numers, nd let Q, 0. For ositive rime integer, the Artin symol Q /Q hs the vlue 1 if Q is the slitting field of in Q, 0 if is rmified
More informationPRIMES AND QUADRATIC RECIPROCITY
PRIMES AND QUADRATIC RECIPROCITY ANGELICA WONG Abstrct We discuss number theory with the ultimte gol of understnding udrtic recirocity We begin by discussing Fermt s Little Theorem, the Chinese Reminder
More informationLECTURE 10: JACOBI SYMBOL
LECTURE 0: JACOBI SYMBOL The Jcobi symbol We wish to generlise the Legendre symbol to ccomodte comosite moduli Definition Let be n odd ositive integer, nd suose tht s, where the i re rime numbers not necessrily
More informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationplays an important role in many fields of mathematics. This sequence has nice number-theoretic properties; for example, E.
Tiwnese J. Mth. 17013, no., 13-143. FIBONACCI NUMBERS MODULO CUBES OF PRIMES Zhi-Wei Sun Dertment of Mthemtics, Nnjing University Nnjing 10093, Peole s Reublic of Chin zwsun@nju.edu.cn htt://mth.nju.edu.cn/
More informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
More informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationUSA Mathematical Talent Search Round 1 Solutions Year 25 Academic Year
1/1/5. Alex is trying to oen lock whose code is sequence tht is three letters long, with ech of the letters being one of A, B or C, ossibly reeted. The lock hs three buttons, lbeled A, B nd C. When the
More information(II.G) PRIME POWER MODULI AND POWER RESIDUES
II.G PRIME POWER MODULI AND POWER RESIDUES I II.C, we used the Chiese Remider Theorem to reduce cogrueces modulo m r i i to cogrueces modulo r i i. For exmles d roblems, we stuck with r i 1 becuse we hd
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationMATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35
MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
More informationLecture 6: Isometry. Table of contents
Mth 348 Fll 017 Lecture 6: Isometry Disclimer. As we hve textook, this lecture note is for guidnce nd sulement only. It should not e relied on when rering for exms. In this lecture we nish the reliminry
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
More informationIntermediate Math Circles Wednesday, November 14, 2018 Finite Automata II. Nickolas Rollick a b b. a b 4
Intermedite Mth Circles Wednesdy, Novemer 14, 2018 Finite Automt II Nickols Rollick nrollick@uwterloo.c Regulr Lnguges Lst time, we were introduced to the ide of DFA (deterministic finite utomton), one
More information(e) if x = y + z and a divides any two of the integers x, y, or z, then a divides the remaining integer
Divisibility In this note we introduce the notion of divisibility for two integers nd b then we discuss the division lgorithm. First we give forml definition nd note some properties of the division opertion.
More informationSome Results on Cubic Residues
Interntionl Journl of Algebr, Vol. 9, 015, no. 5, 45-49 HIKARI Ltd, www.m-hikri.com htt://dx.doi.org/10.1988/ij.015.555 Some Results on Cubic Residues Dilek Nmlı Blıkesir Üniversiresi Fen-Edebiyt Fkültesi
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationTHE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS. Circa 1870, G. Zolotarev observed that the Legendre symbol ( a p
THE QUADRATIC RECIPROCITY LAW OF DUKE-HOPKINS PETE L CLARK Circ 1870, Zolotrev observed tht the Legendre symbol ( p ) cn be interpreted s the sign of multipliction by viewed s permuttion of the set Z/pZ
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationNatural examples of rings are the ring of integers, a ring of polynomials in one variable, the ring
More generlly, we define ring to be non-empty set R hving two binry opertions (we ll think of these s ddition nd multipliction) which is n Abelin group under + (we ll denote the dditive identity by 0),
More informationHomework 4. 0 ε 0. (00) ε 0 ε 0 (00) (11) CS 341: Foundations of Computer Science II Prof. Marvin Nakayama
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 4 1. UsetheproceduredescriedinLemm1.55toconverttheregulrexpression(((00) (11)) 01) into n NFA. Answer: 0 0 1 1 00 0 0 11 1 1 01 0 1 (00)
More information(9) P (x)u + Q(x)u + R(x)u =0
STURM-LIOUVILLE THEORY 7 2. Second order liner ordinry differentil equtions 2.1. Recll some sic results. A second order liner ordinry differentil eqution (ODE) hs the form (9) P (x)u + Q(x)u + R(x)u =0
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected Exercises. g(x) 2 dx 1 2 a
McGill University Mth 354: Honors Anlysis 3 Fll 2012 Assignment 1 Solutions to selected Exercises Exercise 1. (i) Verify the identity for ny two sets of comlex numers { 1,..., n } nd { 1,..., n } ( n )
More informationThe Evaluation Theorem
These notes closely follow the presenttion of the mteril given in Jmes Stewrt s textook Clculus, Concepts nd Contexts (2nd edition) These notes re intended primrily for in-clss presenttion nd should not
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
More information2.4 Linear Inequalities and Interval Notation
.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or
More informationList all of the possible rational roots of each equation. Then find all solutions (both real and imaginary) of the equation. 1.
Mth Anlysis CP WS 4.X- Section 4.-4.4 Review Complete ech question without the use of grphing clcultor.. Compre the mening of the words: roots, zeros nd fctors.. Determine whether - is root of 0. Show
More informationTheory of Computation Regular Languages. (NTU EE) Regular Languages Fall / 38
Theory of Computtion Regulr Lnguges (NTU EE) Regulr Lnguges Fll 2017 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of Finite Automt A finite utomton hs finite set of control
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationarxiv: v9 [math.nt] 8 Jun 2010
Int. J. Number Theory, in ress. ON SOME NEW CONGRUENCES FOR BINOMIAL COEFFICIENTS rxiv:0709.665v9 [mth.nt] 8 Jun 200 Zhi-Wei Sun Roberto Turso 2 Dertment of Mthemtics, Nning University Nning 2009, Peole
More informationParse trees, ambiguity, and Chomsky normal form
Prse trees, miguity, nd Chomsky norml form In this lecture we will discuss few importnt notions connected with contextfree grmmrs, including prse trees, miguity, nd specil form for context-free grmmrs
More information1 Structural induction
Discrete Structures Prelim 2 smple questions Solutions CS2800 Questions selected for Spring 2018 1 Structurl induction 1. We define set S of functions from Z to Z inductively s follows: Rule 1. For ny
More informationOn Arithmetic Functions
Globl ournl of Mthemticl Sciences: Theory nd Prcticl ISSN 0974-00 Volume 5, Number (0, 7- Interntionl Reserch Publiction House htt://wwwirhousecom On Arithmetic Functions Bhbesh Ds Dertment of Mthemtics,
More informationLecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationTheory of Computation Regular Languages
Theory of Computtion Regulr Lnguges Bow-Yw Wng Acdemi Sinic Spring 2012 Bow-Yw Wng (Acdemi Sinic) Regulr Lnguges Spring 2012 1 / 38 Schemtic of Finite Automt control 0 0 1 0 1 1 1 0 Figure: Schemtic of
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationLet S be a numerical semigroup generated by a generalized arithmetic sequence,
Abstrct We give closed form for the ctenry degree of ny element in numericl monoid generted by generlized rithmetic sequence in embedding dimension three. While it is known in generl tht the lrgest nd
More informationIN GAUSSIAN INTEGERS X 3 + Y 3 = Z 3 HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A2 IN GAUSSIAN INTEGERS X + Y = Z HAS ONLY TRIVIAL SOLUTIONS A NEW APPROACH Elis Lmpkis Lmpropoulou (Term), Kiprissi, T.K: 24500,
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationBeginning Darboux Integration, Math 317, Intro to Analysis II
Beginning Droux Integrtion, Mth 317, Intro to Anlysis II Lets strt y rememering how to integrte function over n intervl. (you lerned this in Clculus I, ut mye it didn t stick.) This set of lecture notes
More informationMATH1050 Cauchy-Schwarz Inequality and Triangle Inequality
MATH050 Cuchy-Schwrz Inequlity nd Tringle Inequlity 0 Refer to the Hndout Qudrtic polynomils Definition (Asolute extrem for rel-vlued functions of one rel vrile) Let I e n intervl, nd h : D R e rel-vlued
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationarxiv: v6 [math.nt] 20 Jan 2016
EXPONENTIALLY S-NUMBERS rxiv:50.0594v6 [mth.nt] 20 Jn 206 VLADIMIR SHEVELEV Abstrct. Let S be the set of ll finite or infinite incresing sequences of ositive integers. For sequence S = {sn)},n, from S,
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationMrgolius 2 In the rticulr cse of Ploue's constnt, we tke = 2= 5+i= 5, nd = ;1, then ; C = tn;1 1 2 = ln(2= 5+i= 5) ln(;1) More generlly, we would hve
Ploue's Constnt is Trnscendentl Brr H. Mrgolius Clevelnd Stte University Clevelnd, Ohio 44115.mrgolius@csuohio.edu Astrct Ploue's constnt tn;1 ( 1 2) is trnscendentl. We rove this nd more generl result
More informationMinimal DFA. minimal DFA for L starting from any other
Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More information5. (±±) Λ = fw j w is string of even lengthg [ 00 = f11,00g 7. (11 [ 00)± Λ = fw j w egins with either 11 or 00g 8. (0 [ ffl)1 Λ = 01 Λ [ 1 Λ 9.
Regulr Expressions, Pumping Lemm, Right Liner Grmmrs Ling 106 Mrch 25, 2002 1 Regulr Expressions A regulr expression descries or genertes lnguge: it is kind of shorthnd for listing the memers of lnguge.
More informationk and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationHomework Solution - Set 5 Due: Friday 10/03/08
CE 96 Introduction to the Theory of Computtion ll 2008 Homework olution - et 5 Due: ridy 10/0/08 1. Textook, Pge 86, Exercise 1.21. () 1 2 Add new strt stte nd finl stte. Mke originl finl stte non-finl.
More information#A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z
#A29 INTEGERS 17 (2017) EQUALITY OF DEDEKIND SUMS MODULO 24Z Kurt Girstmir Institut für Mthemtik, Universität Innsruck, Innsruck, Austri kurt.girstmir@uik.c.t Received: 10/4/16, Accepted: 7/3/17, Pulished:
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationMultiplicative functions of polynomial values in short intervals
ACTA ARITHMETICA LXII3 1992 Multilictive functions of olnomil vlues in short intervls b Mohn Nir Glsgow 1 Introduction Let dn denote the divisor function nd let P n be n irreducible olnomil of degree g
More informationOn a Conjecture of Farhi
47 6 Journl of Integer Sequences, Vol. 7 04, Article 4..8 On Conjecture of Frhi Soufine Mezroui, Abdelmlek Azizi, nd M hmmed Zine Lbortoire ACSA Dértement de Mthémtiques et Informtique Université Mohmmed
More informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
More informationdx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.
Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd
More informationON THE EXCEPTIONAL SET IN THE PROBLEM OF DIOPHANTUS AND DAVENPORT
ON THE EXCEPTIONAL SET IN THE PROBLEM OF DIOPHANTUS AND DAVENPORT Andrej Dujell Deprtment of Mthemtics, University of Zgreb, 10000 Zgreb, CROATIA The Greek mthemticin Diophntus of Alexndri noted tht the
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationTHE NUMBER CONCEPT IN GREEK MATHEMATICS SPRING 2009
THE NUMBER CONCEPT IN GREEK MATHEMATICS SPRING 2009 0.1. VII, Definition 1. A unit is tht by virtue of which ech of the things tht exist is clled one. 0.2. VII, Definition 2. A number is multitude composed
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationMath 61CM - Solutions to homework 9
Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationx 2 a mod m. has a solution. Theorem 13.2 (Euler s Criterion). Let p be an odd prime. The congruence x 2 1 mod p,
13. Quadratic Residues We now turn to the question of when a quadratic equation has a solution modulo m. The general quadratic equation looks like ax + bx + c 0 mod m. Assuming that m is odd or that b
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
More informationSurface maps into free groups
Surfce mps into free groups lden Wlker Novemer 10, 2014 Free groups wedge X of two circles: Set F = π 1 (X ) =,. We write cpitl letters for inverse, so = 1. e.g. () 1 = Commuttors Let x nd y e loops. The
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors Pre-Chpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationIntroduction to Olympiad Inequalities
Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................
More informationMathematics Number: Logarithms
plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement
More informationCMPSCI 250: Introduction to Computation. Lecture #31: What DFA s Can and Can t Do David Mix Barrington 9 April 2014
CMPSCI 250: Introduction to Computtion Lecture #31: Wht DFA s Cn nd Cn t Do Dvid Mix Brrington 9 April 2014 Wht DFA s Cn nd Cn t Do Deterministic Finite Automt Forml Definition of DFA s Exmples of DFA
More informationGeometric Sequences. Geometric Sequence a sequence whose consecutive terms have a common ratio.
Geometric Sequences Geometric Sequence sequence whose consecutive terms hve common rtio. Geometric Sequence A sequence is geometric if the rtios of consecutive terms re the sme. 2 3 4... 2 3 The number
More informationBoolean Algebra. Boolean Algebra
Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
More informationName Ima Sample ASU ID
Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to
More informationMATH342 Practice Exam
MATH342 Practice Exam This exam is intended to be in a similar style to the examination in May/June 2012. It is not imlied that all questions on the real examination will follow the content of the ractice
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information1. For each of the following theorems, give a two or three sentence sketch of how the proof goes or why it is not true.
York University CSE 2 Unit 3. DFA Clsses Converting etween DFA, NFA, Regulr Expressions, nd Extended Regulr Expressions Instructor: Jeff Edmonds Don t chet y looking t these nswers premturely.. For ech
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 1-8, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationHarvard University Computer Science 121 Midterm October 23, 2012
Hrvrd University Computer Science 121 Midterm Octoer 23, 2012 This is closed-ook exmintion. You my use ny result from lecture, Sipser, prolem sets, or section, s long s you quote it clerly. The lphet is
More information