# MTH 505: Number Theory Spring 2017

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1 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 s some vlue tht we wnt to py. The eqution hs solution px, yq P N 2 if nd only if we cn mke chnge for \$c, nd in this cse we sy tht c is p, q-representle. More generlly, we will consider the set of p, q-representtions of c: R,,c : tpx, yq P N 2 : x ` y cu. The prolem is trivil when 0 so we will lwys ssume tht 0, i.e., tht nd re oth nonzero. 2.. Consider nturl numers,, c P N with d gcdp, q, where d nd d. () If d c prove tht R,,c H. () If d c with c dc prove tht R,,c R,,c. [Unlike the cse of Diophntine equtions, it is possile tht oth of these sets could e empty.] Proof. (): Let d c nd ssume for contrdiction tht R,,c is not empty, i.e., ssume tht there exists pir of nturl numers px, yq P N 2 such tht x ` y c. But then we hve which contrdicts the fct tht d c. c x ` y pd qx ` pd q dp x ` yq, (): Now suppose tht d c so tht c dc for some c P Z. Since c nd d re oth positive we must hve c P N. To show tht R,,c Ď R,,c consider ny px, yq P N 2, so tht x` y c. Then we hve x ` y c dp x ` yq dpc q pd qx ` pd qy pdc q x ` y c, which sys tht px, yq P R,,c s desired. Conversely, consider ny px, yq P R,,c, so tht x ` y c. Then we hve x ` y c pd qx ` pd qy pdc q dp x ` yq dpc q x ` y c, which sys tht px, yq P R,,c. (The lst step used multiplictive cncelltion.)

2 The previous result llows us to restrict our ttention to coprime nd Let,, c P N with 0 nd gcdp, q. If R,,c H (i.e., if c is p, q-representle) prove tht there exists unique representtion pu, vq P R,,c with the property 0 ď u ă. [Hint: For existence, let px, yq P R,,c e n ritrry solution. Since 0 there exists quotient nd reminder of x mod. For uniqueness, use the coprimlity of nd to pply Euclid s Lemm.] Proof. If R,,c H then there exists some pir px, yq P N 2 such tht x ` y c. Since 0 there exists pir of integers q, r P Z such tht " x q ` r Then sustituting x q ` r gives 0 ď r ă x x ` y c pq ` rq ` y c r ` pq ` yq c. It only remins to check tht pu, vq : pr, q ` yq P N 2 nd we lredy know tht r P N. Since r ă x we lso hve q px rq ą 0, which since ą 0 implies tht q ą 0. But then since y P N we hve q ` y P N s desired. This proves existence. For uniqueness, ssume tht we hve pu, v q nd pu 2, v 2 q in R,,c with 0 ď u ă nd 0 ď u 2 ă. Then since u ` v c u 2 ` v 2 we see tht u ` v u 2 ` v 2 pu u 2 q pv 2 v q, which implies tht divides pu u 2 q. But then since gcdp, q, Euclid s Lemm sys tht pu u 2 q. If pu u 2 q 0 then we re done. Otherwise, suppose without loss of generlity tht u u 2 ą 0. Then the fct tht pu u 2 q implies tht ď u u 2 ď u which contrdicts the fct tht u ă. This contrdiction shows tht pu u 2 q 0 nd then the eqution pv 2 v q pu u 2 q 0 0 together with the fct 0 implies tht pv 2 v q 0 s desired Let, P N e coprime with 0. If c p q prove tht R,,c H. Tht is, prove tht the numer p q is not p, q-representle. [Hint: Let c p q nd ssume for contrdiction there exists representtion px, yq P R,,c. Show tht the cses x ă nd x ě oth led to the contrdiction y ă 0. You cn use 2.2 for the cse x ă.] Proof. Assume for contrdiction tht we hve representtion x ` y p q with px, yq P N 2. From 2.2 this implies tht there exists representtion u ` v p q

3 with pu, vq P N 2 nd 0 ď u ă. Now oserve tht u ` v u ` v pu ` q p vq. The lst eqution sys tht divides pu ` q nd then since nd re coprime we otin pu ` q from Euclid s Lemm. Since u ` ą 0 this implies tht ď u ` [this rgument is in the notes] ut we lredy know tht u ă (i.e., u ` ď ) so we conclude tht u `. Finlly, we sustitute u to otin which contrdicts the fct tht v P N. u ` v p q ` v ` v v v, [Sorry I didn t follow my own hint very closely.] 2.4. Let, P N e coprime with 0. In this exercise you will prove y induction tht every numer c ą p q is p, q-representle. () Prove the result when or. () From now on we will ssume tht ě 2 nd ě 2. In this cse prove tht the numer p ` q is p, q-representle. [Hint: From the Eucliden Algorithm nd 2.2 there exist x, y P Z with x ` y nd 0 ď x ă. Prove tht px q P N nd py ` q P N, nd hence is vlid representtion.] px q ` py ` q p ` q (c) Let n ě p ` q nd ssume for induction tht n is p, q-representle. In this cse prove tht n ` is lso p, q-representle. [Hint: Let x, y e s in prt (). By the induction hypothesis nd 2.2 there exist x, y P N with x ` y n nd 0 ď x ă. Note tht px ` x q ` py ` y q pn ` q. If y ` y ě 0 then you re done. Otherwise, show tht is vlid representtion.] px ` x q ` py ` y ` q pn ` q Proof. (): Since the prolem is symmetric in nd we will ssume without loss of generlity tht. Now we wnt to show tht every numer c ą p q, i.e., every numer c ě 0 is p, q-representle. But this is certinly true ecuse p0q ` p0q 0 is vlid representtion of c 0 nd pq ` pc q c is vlid representtion of c ą 0. This solves the prolem when or so from now on we will ssume tht ě 2 nd ě 2. (): Bse Cse. Since gcdp, q the Eucliden Algorithm gives integers x, y P Z such tht x ` y nd from 2.2 we cn ssume tht 0 ď x ă. [Actully this is it esier

4 thn 2.2 ecuse we don t require y ě 0.] If x 0 then we hve y x ` y which implies tht, contrdicting the fct tht ě 2. Thus we must hve x ě, i.e., x P N. To complete the proof, ssume for contrdiction tht py ` q ă 0, i.e., y ` ď 0. This implies tht y ď nd hence y ď. Finlly, since px q ă 0 we otin the desired contrdiction: x ` y ď x px q ă 0. We conclude tht px q nd py ` q re nturl numers, so px q ` py ` q px ` y q ` ` is vlid p, q-representiton of p ` q. (c): Induction Step. Let n ě p ` q nd ssume for induction tht there exist nturl numers px, yq P N 2 such tht x ` y n. In this cse we wnt to show tht n ` is lso p, q-representle. To do this, recll from prt () tht we hve integers x, y P Z with the following properties: x ` y, ď x ď, y ` ě. Now dd the equtions x ` y n nd x ` y to otin px ` x q ` py ` y q n `, where x ` x ě 0. If we lso hve y ` y ě 0 then we re done, so ssume tht y ` y ă 0. Since y ` ě nd y ě 0 we hve py ` y ` q ě. It only remins to check tht px ` x q ě 0. To see this we use the ssumptions pn ` q ě p ` 2q nd py ` y ` q ď 0 to otin n ` px ` x q ` py ` y q ą ` 2 px ` x q ě py ` y q ` 2 ą py ` y ` q ` 2 ě p0q ` 2 ą p q ą 0. By cncelling ą 0 from oth sides of px ` x q ą p q we otin px ` x q ą p q nd hence px ` x q ě 0 s desired. It follows tht px ` x q ` py ` y ` q px ` yq ` px ` y q ` p ` q n ` ` 0 is vlid p, q-representtion of n `. [Tht ws tricky.] Let, P N e coprime with 0. So fr you hve proved tht R,,p q 0 nd R,,c ě for ll c ą p q. The next prolem gives rough lower ound for the totl numer of p, q-representtions.

5 2.5. Let, P N e coprime with 0. Prove tht Y c R,,c ě mxtn P N : n ď c{pqu. ] [Hint: We know from clss tht the integer solutions of x ` y c hve the form px, yq pcx k, cy ` P Z, where x, y P Z re some specific integers stisfying x `y. Now prove tht the nturl numer solutions correspond to vlues of k P Z such tht cy ď k ď cx. Counting these integers is delicte ut you should e le to give rough ound.] Proof. Consider,, c P N with gcdp, q. From 2.2 there exist integers x, y P Z such tht x ` y nd 0 ď x ă. We know from clss tht the complete integer solution to the eqution x ` y c is given y px, yq pcx k, cy ` P Z, nd our jo is to discover which of these solutions re non-negtive. Tht is, we need to find ll integers k P Z such tht the following two inequlities hold: cx k ě 0 cy ` k ě 0. These two inequlities cn e written in terms of frctions to otin cy ď k ď cx. Ech such vlue of k P Z corresponds to non-negtive solution of x`y c, so we conclude tht R,,c is equl to the numer of integers in the closed rel numer intervl r cy {, cx {s. The exct count is tricky, ut the floor of the length of the intervl provides lower ound: Z ^ cx R,,c ě cy Z cx ` cy ^ Z cpx ` y ^ q Y c ]. Unfortuntely this rough ound gives us no informtion when c ă, i.e., when tc{pqu 0. With it more work one could compute the exct formul: for ny x ` y we hve ( ) R,,c c " " cy cx `, where txu : x txu is the frctionl prt of the rtionl numer x P Q. This formul is due to Tieriu Popoviciu in 953.

6 2.6. Let, P N e coprime with 0. Given n integer 0 ă c ă such tht c nd c, use Popoviciu s formul ( ) to show tht R,,c ` R,,p cq. [Hint: Use the fct tht t xu txu when x R Z.] Proof. Consider,, c P N with gcdp, q, 0 ă c ă, nd where nd do not divide c. Then for ny integers x ` y Popoviciu s formul gives R,,p cq c " " p cqy p cqx ` 2 c "y cy "x cx. But now oserve tht for ll integers n P Z nd non-integer rtionls x P Q we hve tn xu t xu txu. Thus the ove formul ecomes R,,p cq 2 c "y cy "x cx 2 c ˆ " ˆ " cy cx ˆ " " c cy cx ` R,,c. In conclusion, one cn show from 2.6 tht there exist exctly p qp q 2 nturl numers tht re not p, q-representle. This fct ws first proved y Jmes Joseph Sylvester in 884. Proof. I didn t sk you to show this, ut here s the proof. Let gcdp, q. Then we know tht every integer c ě is p, q-representle. [In fct we know tht every integer c ą p q is representle, ut we don t need this right now.] Of the ` elements of the set tc P Z : 0 ď c ď u we know tht elements re multiples of, nd elements re multiples of. Furthermore, since gcdp, q we know tht the only elements tht re multiples of oth nd re 0 nd. We conclude tht there re exctly p ` q p ` 2q p ` q p qp q elements of the set tht re not multiple of or. The result of Prolem 2.6 sys tht exctly hlf of these numers re p, q-representle. Epilogue: The proofs ove re lgeric, ut there is lso eutiful geometric wy to think out the Froenius coin prolem. Consider, P N with 0 nd gcdp, q. Lel ech point px, yq P Z 2 of the integer lttice y the numer x ` y. Note tht points on the sme line of slope { receve the sme lel. The prolem is to count the integer points on the line x ` y c tht lie in the first qudrnt. For exmple, here is the lelling corresponding to the coprime pir p, q p5, 8q:

7 I hve drwn the lines 5x ` 8y nd 5x ` 8y 0. It ws reltively esy to show tht every lel c ě 40 occurs in the first qudrnt, ut the numers elow 40 re more tricky. I hve outlined the numers elow 40 re re not multiples of 5 or 8 ut re still p5, 8q-representle. We oserve tht there re p5 qp8 q{2 4 such numers, s expected. I hve lso outlined the numers in the fourth qudrnt tht re not p5, 8q-representle. Oserve tht these two shpes re congruent up to 80 rottion, nd in fct this is the trnsformtion c ÞÑ p cq. Oserve further tht the two shpes fit together perfectly to mke n p q ˆ p q rectngle. This is the geometric explntion for Sylvester s formul p qp q. 2

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### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2. Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O 1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Math 154B Elementary Algebra-2 nd Half Spring 2015 Mth 154B Elementry Alger- nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know

### Mathematics. Area under Curve. Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding

### Intermediate 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

### Riemann Sums and Riemann Integrals Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties

### Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

### Lesson 25: Adding and Subtracting Rational Expressions Lesson 2: Adding nd Subtrcting Rtionl Expressions Student Outcomes Students perform ddition nd subtrction of rtionl expressions. Lesson Notes This lesson reviews ddition nd subtrction of frctions using

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### 1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check

### Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

### Section 6.1 INTRO to LAPLACE TRANSFORMS Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### AP Calculus AB Summer Packet AP Clculus AB Summer Pcket Nme: Welcome to AP Clculus AB! Congrtultions! You hve mde it to one of the most dvnced mth course in high school! It s quite n ccomplishment nd you should e proud of yourself

### dx 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

### SUMMER 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

### Minimal 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

### 5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### Surface 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

### Natural 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),

### Section 7.1 Area of a Region Between Two Curves Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers. 8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Section 6.1 Definite Integral Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Homework 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}.

### 1. 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

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

### Linear Inequalities. Work Sheet 1 Work Sheet 1 Liner Inequlities Rent--Hep, cr rentl compny,chrges \$ 15 per week plus \$ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend

### ARITHMETIC 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

### MATH 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.

### 10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### The 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

### Math 259 Winter Solutions to Homework #9 Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

### I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

### Properties of the Riemann Integral Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

### 20 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

### List 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

### IN 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,

### Lecture 2: Fields, Formally Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of

### Torsion in Groups of Integral Triangles Advnces in Pure Mthemtics, 01,, 116-10 http://dxdoiorg/1046/pm011015 Pulished Online Jnury 01 (http://wwwscirporg/journl/pm) Torsion in Groups of Integrl Tringles Will Murry Deprtment of Mthemtics nd Sttistics,

### Is there an easy way to find examples of such triples? Why yes! Just look at an ordinary multiplication table to find them! PUSHING PYTHAGORAS 009 Jmes Tnton A triple of integers ( bc,, ) is clled Pythgoren triple if exmple, some clssic triples re ( 3,4,5 ), ( 5,1,13 ), ( ) fond of ( 0,1,9 ) nd ( 119,10,169 ). + b = c. For

### State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response 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 > Section 3. Mximum Principle nd Uniqueness Let u (x; y) e smooth solution in. Then the mximum vlue exists nd is nite. (x ; y ) ; i.e., M mx fu (x; y) j (x; y) in g Furthermore, this vlue cn e otined y point