Jacobi s four squares identity Martin Klazar


 Leonard Skinner
 3 years ago
 Views:
Transcription
1 Jacobi s four squares identity Martin Klazar (lecture on the 7th PhD conference) Ostrava, September 10, 013 C. Jacobi [] in 189 proved that for any integer n 1, r (n) = #{(x 1, x, x 3, x ) Z ( i=1 x i = n} = 8 d ) d. d n In other words, the number of ways to express n as a sum of four squares of integers equals eight times the sum of the divisors of n not divisible by four. Thus r (n) = 8(1 +...) 8 for every n 1, which gives as a corollary Lagrange s theorem from 1770 that every natural number is a sum of four squares. We give a complete and purely formal proof of Jacobi s identity by generating functions; the proof is due to Hirschhorn [1] in Idea: since r g (n), the number of expressions of an integer n 0 as a sum of g squares of integers, equals to the coefficient of x n in (the formal power series expansion of) ( + x n) g, n= and d n d equals to the coefficient of xn in the Lambert series k, nx kn = nx n 1 x n, d n it suffices to derive the identity ( x n ) = ( nx n 1 x nx n ) n 1 x n (1) Jacobi s identity in the GF language. (We write briefly for + n= and for.) We achieve it by a succession of the next seven identities. 1
2 1. ( 1) n x n = 1 x n 1 + x n.. JTI: (1 x n )(1 + zx n 1 )(1 + z 1 x n 1 ) = z n x n. 3. (z z 1 ) (1 x n )(1 z x n )(1 z x n ) = ( 1) n z n+1 x n(n+1)/.. (1 x n ) 3 = 1 (n + 1)( 1) n x n(n+1)/. 5. (1 x n ) 6 = 1 ((r + 1) (s) )x r +r+s. r,s 6. (1 x n ) 6 = Q n [ 1 8 ( (n 1)x n x n 1 nxn 1 + x n )], where Q n = (1 x n ) (1 + x n ) (1 + x n 1 ). 7. (1 + x n ) (1 x n ) = Q n. The key identity, from which we derive everything, is, the celebrated Jacobi s triple product identity (JTI for short), obtained by C. Jacobi in []. We first deduce (1) from the identities, then derive the seven identities assuming the JTI, and at the end we give a proof for the JTI. We have that ( ( 1) n x n) (id.1) = ( 1 x n ) (the left side of id. 6) = 1 + x n (the left side of id. 7) equals to the ratio of the right sides 1 8 ( (n 1)x n 1 ) nxn. 1 + x n x n
3 Replacing x with x, we obtain the identity ( x n ) = ( (n 1)x n 1 ) + nxn, 1 x n x n which is similar to (1). We rewrite the last sum as ( (n 1)x n 1 ) + nxn 1 x n 1 1 x n ( nx n ) nxn. 1 xn 1 + x n The first sum equals nx n by regrouping, and the summand in the 1 x n second sum equals, by subtracting the two fractions, to nxn. Thus the 1 x n obtained identity is not just similar to but indeed identical to (1). We start from identity, the JTI, and derive from it the rest. 1. We set z = 1 in the JTI and get that ( 1) n x n equals to (1 x n )(1 x n 1 ). Since (1 x n )(1 x n 1 ) = (1 x n ) and (1 x n ) = (1 x n )/(1 + x n ), regrouping yields the right side of We set z = xz in the JTI, then x = x 1/ and multiply the result by z. 3. We differentiate 3 by z, set z = 1 and multiply the result by Squaring gives (1 x n ) 6 = 1 (m + 1)(n + 1)( 1) m+n x (m +m+n +n)/. m,n We split the sum in two subsums, the first with even m + n and the second with odd m + n. In the first subsum we change the variables to r = (m + n)/, s = (m n)/, and in the second to r = (m n 1)/, s = (m+n+1)/. It is easy to check that in both cases m +m+n +n turns into (r +r +s ) and ( 1) m+n (m + 1)(n + 1) into (r + 1) (s). Hence the two subsums coincide and the sign disappears; we get the right side of 5. 5, 6. We write the right side of 5 as a difference of two sums, by the difference (r + 1) (s), and separate in each sum the variables r and s. Then we express the coefficients by differentiation and obtain 1 ( s x s (1 + x d dx ) r x r +r r 3 x r +r x d dx x s). s
4 We replace each of the four s by a using the JTI, setting z = 1 in s and z = x in r ( 1 gets cancelled by 1 + x0 ): (1 x n )(1 + x n 1 ) (1 + x d dx ) (1 x n )(1 + x n ) minus (1 x n )(1 + x n ) x d dx (1 x n )(1 + x n 1 ). We differentiate the infinite products by means of the identity ( f n ) = fn f n/f n and, denoting a = 1 + x n, b = 1 + x n 1, c = 1 x n and taking out the factor Q n = (abc), get the expression Qn [ 1 + x ( (a c) a c (b c) )]. b c The last summand simplifies after an easy calculation to (a /a b /b). We get the right side of We have Qn = (1 x n ) (1 + x n ) (1 + x n 1 ). Since 1 x n = (1 x n )(1 + x n ) and (1 + x n )(1 + x n 1 ) = (1 + x n ), regrouping gives the left side of 7. It remains to prove the JTI (1 x n )(1 + zx n 1 )(1 + z 1 x n 1 ) = z n x n. We give a purely formal proof. Let S(z, x) be the product on the left side. It follows that zx S(zx, x) = S(z, x), because the substitution z zx almost preserves the two arithmetic progressions of odd positive integers in the two exponents of x: it just adds to S(z, x) the factor (1 + z 1 x 1 )/(1 + zx) = 1/zx. Therefore if we expand S(z, x) in the integral powers of z as S(z, x) = a n z n = a n (x)z n (n runs through the whole Z) with the power series coefficients a n C[[x]], comparison of the coefficients of z n on both sides of the functional equation gives the relation x n 1 a n 1 = a n, n Z.
5 Thus a 1 = xa 0 = a 1 (n = 1, 0), a = x 3 a 1 = x 3 a 1 = a (n =, 1) and a = x a 0 = a. In general, a n = a n = x n a 0, n = 1,,..., since (n 1) = n. (The equality a n = a n is immediate also from the symmetry S(z, x) = S(z 1, x).) So we have deduced that S(z, x) = (1 x n )(1 + zx n 1 )(1 + z 1 x n 1 ) = a 0 (x) z n x n and are almost done it only remains to be shown that a 0 = 1. This we prove by three specializations of the last identity (the JTI with the undetermined coefficient a 0 (x)): (i) z = x = 0, (ii) z = i and (iii) z = 1, x = x. Then (i) gives that a 0 (0) = 1, (ii) gives that (1 x n )(1 + x n ) = a 0 (x) ( 1) n x (n) and (iii) gives that (1 x 8n )(1 x 8n ) = a 0 (x ) ( 1) n x n. Clearly, the sums on the right sides are equal. So are the products on the left sides, despite appearance: since (1 x 8n )(1 x 8n ) = (1 x n ), 1 x 8n = (1 x n )(1 + x n ) and (1 x n )(1 x n ) = (1 x n ), the product of (iii) equals to that of (ii). Hence the remaining factors have to be equal too, a 0 (x) = a 0 (x ), which forces that a 0 (x) = a 0 (0) = 1, as we needed to show. This completes the proof of the JTI and of Jacobi s four squares identity. Exercise. Justify rigorously formal manipulations in the above proof. In particular, clarify in what sense does the formal equality (1 x n )(1 + zx n 1 )(1 + z 1 x n 1 ) = a n (x)z n hold and why it is preserved by the substitution z zx. References [1] M. D. Hirschhorn, A simple proof of Jacobi s foursquare theorem, Proc. Amer. Math. Soc. 101 (1987), [] C. G. J. Jacobi, Fundamenta nova theoriae functionum ellipticarum, Sumtibus fratrum,
Taylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationMath 115 Spring 2014 Written Homework 3 Due Wednesday, February 19
Math 11 Spring 01 Written Homework 3 Due Wednesday, February 19 Instructions: Write complete solutions on separate paper (not spiral bound). If multiple pieces of paper are used, they must be stapled with
More informationSome facts about polynomials modulo m (Full proof of the Fingerprinting Theorem)
Some facts about polynomials modulo m (Full proof of the Fingerprinting Theorem) In order to understand the details of the Fingerprinting Theorem on fingerprints of different texts from Chapter 19 of the
More informationWorksheet on induction Calculus I Fall 2006 First, let us explain the use of for summation. The notation
Worksheet on induction MA113 Calculus I Fall 2006 First, let us explain the use of for summation. The notation f(k) means to evaluate the function f(k) at k = 1, 2,..., n and add up the results. In other
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationChapter 7. Induction and Recursion. Part 1. Mathematical Induction
Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply
More informationCHAPTER 2 FOURIER SERIES
CHAPTER 2 FOURIER SERIES PERIODIC FUNCTIONS A function is said to have a period T if for all x,, where T is a positive constant. The least value of T>0 is called the period of. EXAMPLES We know that =
More informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationTRIANGLES ON THE LATTICE OF INTEGERS. Department of Mathematics Rowan University Glassboro, NJ Andrew Roibal and Abdulkadir Hassen
TRIANGLES ON THE LATTICE OF INTEGERS Andrew Roibal and Abdulkadir Hassen Department of Mathematics Rowan University Glassboro, NJ 08028 I. Introduction In this article we will be studying triangles whose
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More information2.6 Exponents and Order of Operations
2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationMathematical Induction
Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers
More informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More informationAlgebra for Digital Communication
EPFL  Section de Mathématiques Algebra for Digital Communication Fall semester 2008 Solutions for exercise sheet 1 Exercise 1. i) We will do a proof by contradiction. Suppose 2 a 2 but 2 a. We will obtain
More informationContinued fractions and good approximations.
Continued fractions and good approximations We will study how to find good approximations for important real life constants A good approximation must be both accurate and easy to use For instance, our
More informationIntegrals of Rational Functions
Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t
More information1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style
Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with
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 information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationSome Notes on Taylor Polynomials and Taylor Series
Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited
More informationThis is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0).
This is Radical Expressions and Equations, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons byncsa 3.0 (http://creativecommons.org/licenses/byncsa/
More information2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2
00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the
More informationThe Geometric Series
The Geometric Series Professor Jeff Stuart Pacific Lutheran University c 8 The Geometric Series Finite Number of Summands The geometric series is a sum in which each summand is obtained as a common multiple
More informationProof: A logical argument establishing the truth of the theorem given the truth of the axioms and any previously proven theorems.
Math 232  Discrete Math 2.1 Direct Proofs and Counterexamples Notes Axiom: Proposition that is assumed to be true. Proof: A logical argument establishing the truth of the theorem given the truth of the
More information3. Power of a Product: Separate letters, distribute to the exponents and the bases
Chapter 5 : Polynomials and Polynomial Functions 5.1 Properties of Exponents Rules: 1. Product of Powers: Add the exponents, base stays the same 2. Power of Power: Multiply exponents, bases stay the same
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More information8.7 Mathematical Induction
8.7. MATHEMATICAL INDUCTION 8135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture
More informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationMATH Fundamental Mathematics II.
MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/funmath2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics
More informationPROOFS BY DESCENT KEITH CONRAD
PROOFS BY DESCENT KEITH CONRAD As ordinary methods, such as are found in the books, are inadequate to proving such difficult propositions, I discovered at last a most singular method... that I called the
More informationLagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given.
Polynomials (Ch.1) Study Guide by BS, JL, AZ, CC, SH, HL Lagrange Interpolation is a method of fitting an equation to a set of points that functions well when there are few points given. Sasha s method
More informationLESSON 1 PRIME NUMBERS AND FACTORISATION
LESSON 1 PRIME NUMBERS AND FACTORISATION 1.1 FACTORS: The natural numbers are the numbers 1,, 3, 4,. The integers are the naturals numbers together with 0 and the negative integers. That is the integers
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationArithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get
Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real
More informationGREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
More informationSolve Systems of Equations by the Addition/Elimination Method
Solve Systems of Equations by the Addition/Elimination Method When solving systems, we have found that graphing is very limited. We then considered the substitution method which has its limitations as
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More information= = = = = = =2 3 10) = =1 2 12) = )
11 ) 11 11  1 1 1 1 1 19  1 10 11 91 11 111 10 91     1 11 1   1 11 1 91   1   1   11 1 11  110 9 1   1 1.. 1. 1... 10. 9. 9 10. 1 11. 1. 1. 1. 10 1. 1 1 1 110 9 1 9 0
More informationMATH REVIEW KIT. Reproduced with permission of the Certified General Accountant Association of Canada.
MATH REVIEW KIT Reproduced with permission of the Certified General Accountant Association of Canada. Copyright 00 by the Certified General Accountant Association of Canada and the UBC Real Estate Division.
More informationExam #5 Sequences and Series
College of the Redwoods Mathematics Department Math 30 College Algebra Exam #5 Sequences and Series David Arnold Don Hickethier Copyright c 000 DonHickethier@Eureka.redwoods.cc.ca.us Last Revision Date:
More information15. Symmetric polynomials
15. Symmetric polynomials 15.1 The theorem 15.2 First examples 15.3 A variant: discriminants 1. The theorem Let S n be the group of permutations of {1,, n}, also called the symmetric group on n things.
More informationManipulating Power Series
LECTURE 24 Manipulating Power Series Our technique for solving di erential equations by power series will essentially be to substitute a generic power series expression y(x) a n (x x o ) n into a di erential
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationGenerating Functions
Generating Functions If you take f(x =/( x x 2 and expand it as a power series by long division, say, then you get f(x =/( x x 2 =+x+2x 2 +x +5x 4 +8x 5 +x 6 +. It certainly seems as though the coefficient
More informationProblem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB
Problem Set 5 1. (a) Fourdigit number S = aabb is a square. Find it; (hint: 11 is a factor of S) (b) If n is a sum of two square, so is 2n. (Frank) Solution: (a) Since (A + B) (A + B) = 0, and 11 0, 11
More information12 Greatest Common Divisors. The Euclidean Algorithm
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to
More information1.3 Polynomials and Factoring
1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.
More information3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style
Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or
More informationAdvanced Higher Mathematics Course Assessment Specification (C747 77)
Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational
More informationSan Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS
San Jose Math Circle October 17, 2009 ARITHMETIC AND GEOMETRIC PROGRESSIONS DEFINITION. An arithmetic progression is a (finite or infinite) sequence of numbers with the property that the difference between
More informationMath 115 Spring 2011 Written Homework 5 Solutions
. Evaluate each series. a) 4 7 0... 55 Math 5 Spring 0 Written Homework 5 Solutions Solution: We note that the associated sequence, 4, 7, 0,..., 55 appears to be an arithmetic sequence. If the sequence
More informationComputing divisors and common multiples of quasilinear ordinary differential equations
Computing divisors and common multiples of quasilinear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univlille1.fr
More informationEULER S THEOREM. 1. Introduction Fermat s little theorem is an important property of integers to a prime modulus. a p 1 1 mod p.
EULER S THEOREM KEITH CONRAD. Introduction Fermat s little theorem is an important property of integers to a prime modulus. Theorem. (Fermat). For prime p and any a Z such that a 0 mod p, a p mod p. If
More informationSometimes it is easier to leave a number written as an exponent. For example, it is much easier to write
4.0 Exponent Property Review First let s start with a review of what exponents are. Recall that 3 means taking four 3 s and multiplying them together. So we know that 3 3 3 3 381. You might also recall
More informationP.E.R.T. Math Study Guide
A guide to help you prepare for the Math subtest of Florida s Postsecondary Education Readiness Test or P.E.R.T. P.E.R.T. Math Study Guide www.perttest.com PERT  A Math Study Guide 1. Linear Equations
More informationFlorida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper
Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies  Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationThe cyclotomic polynomials
The cyclotomic polynomials Notes by G.J.O. Jameson 1. The definition and general results We use the notation e(t) = e 2πit. Note that e(n) = 1 for integers n, e(s + t) = e(s)e(t) for all s, t. e( 1 ) =
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More informationSection 4.1 Rules of Exponents
Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells
More informationUnit 3 Polynomials Study Guide
Unit Polynomials Study Guide 75 Polynomials Part 1: Classifying Polynomials by Terms Some polynomials have specific names based upon the number of terms they have: # of Terms Name 1 Monomial Binomial
More informationCM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation
CM2202: Scientific Computing and Multimedia Applications General Maths: 2. Algebra  Factorisation Prof. David Marshall School of Computer Science & Informatics Factorisation Factorisation is a way of
More information61. REARRANGEMENTS 119
61. REARRANGEMENTS 119 61. Rearrangements Here the difference between conditionally and absolutely convergent series is further refined through the concept of rearrangement. Definition 15. (Rearrangement)
More informationx if x 0, x if x < 0.
Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the
More informationPolynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if
1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series
ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don
More informationDiscrete Mathematics: Homework 7 solution. Due: 2011.6.03
EE 2060 Discrete Mathematics spring 2011 Discrete Mathematics: Homework 7 solution Due: 2011.6.03 1. Let a n = 2 n + 5 3 n for n = 0, 1, 2,... (a) (2%) Find a 0, a 1, a 2, a 3 and a 4. (b) (2%) Show that
More informationMATH 65 NOTEBOOK CERTIFICATIONS
MATH 65 NOTEBOOK CERTIFICATIONS Review Material from Math 60 2.5 4.3 4.4a Chapter #8: Systems of Linear Equations 8.1 8.2 8.3 Chapter #5: Exponents and Polynomials 5.1 5.2a 5.2b 5.3 5.4 5.5 5.6a 5.7a 1
More informationContinued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationAlgebra 1A and 1B Summer Packet
Algebra 1A and 1B Summer Packet Name: Calculators are not allowed on the summer math packet. This packet is due the first week of school and will be counted as a grade. You will also be tested over the
More informationMath Help and Additional Practice Websites
Name: Math Help and Additional Practice Websites http://www.coolmath.com www.aplusmath.com/ http://www.mathplayground.com/games.html http://www.ixl.com/math/grade7 http://www.softschools.com/grades/6th_and_7th.jsp
More informationSOLUTIONS FOR PROBLEM SET 2
SOLUTIONS FOR PROBLEM SET 2 A: There exist primes p such that p+6k is also prime for k = 1,2 and 3. One such prime is p = 11. Another such prime is p = 41. Prove that there exists exactly one prime p such
More informationHFCC Math Lab Arithmetic  4. Addition, Subtraction, Multiplication and Division of Mixed Numbers
HFCC Math Lab Arithmetic  Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers.
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationPEN A Problems. a 2 +b 2 ab+1. 0 < a 2 +b 2 abc c, x 2 +y 2 +1 xy
Divisibility Theory 1 Show that if x,y,z are positive integers, then (xy + 1)(yz + 1)(zx + 1) is a perfect square if and only if xy +1, yz +1, zx+1 are all perfect squares. 2 Find infinitely many triples
More informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationSECTION 1.5: PIECEWISEDEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS
(Section.5: PiecewiseDefined Functions; Limits and Continuity in Calculus).5. SECTION.5: PIECEWISEDEFINED FUNCTIONS; LIMITS AND CONTINUITY IN CALCULUS LEARNING OBJECTIVES Know how to evaluate and graph
More informationNo Solution Equations Let s look at the following equation: 2 +3=2 +7
5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2016 47 4. Diophantine Equations A Diophantine Equation is simply an equation in one or more variables for which integer (or sometimes rational) solutions
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION
ARITHMETICAL FUNCTIONS II: CONVOLUTION AND INVERSION PETE L. CLARK 1. Sums over divisors, convolution and Möbius Inversion The proof of the multiplicativity of the functions σ k, easy though it was, actually
More informationSequences and Series
Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will
More informationConcentration of points on two and three dimensional modular hyperbolas and applications
Concentration of points on two and three dimensional modular hyperbolas and applications J. Cilleruelo Instituto de Ciencias Matemáticas (CSICUAMUC3MUCM) and Departamento de Matemáticas Universidad
More information= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationCHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY
January 10, 2010 CHAPTER SIX IRREDUCIBILITY AND FACTORIZATION 1. BASIC DIVISIBILITY THEORY The set of polynomials over a field F is a ring, whose structure shares with the ring of integers many characteristics.
More informationArithmetic and Geometric Sequences
Arithmetic and Geometric Sequences Felix Lazebnik This collection of problems is for those who wish to learn about arithmetic and geometric sequences, or to those who wish to improve their understanding
More informationPartial Fractions. p(x) q(x)
Partial Fractions Introduction to Partial Fractions Given a rational function of the form p(x) q(x) where the degree of p(x) is less than the degree of q(x), the method of partial fractions seeks to break
More informationAn Introduction to Galois Fields and ReedSolomon Coding
An Introduction to Galois Fields and ReedSolomon Coding James Westall James Martin School of Computing Clemson University Clemson, SC 296341906 October 4, 2010 1 Fields A field is a set of elements on
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationSample Problems. Lecture Notes Similar Triangles page 1
Lecture Notes Similar Triangles page 1 Sample Problems 1. The triangles shown below are similar. Find the exact values of a and b shown on the picture below. 2. Consider the picture shown below. a) Use
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More information