CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

Size: px
Start display at page:

Download "CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12"

Transcription

1 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. I would like to thank my mentor Avan for introducing and guiding me through this extremely interesting material. I would like to cite Steuding s detailed but slightly flawed book as the main source of learning and Andreescu and Andrica s book as an inspiration for numerous fun experiments I have made this summer. Contents. Continued Fractions 2. Solution to Pell s Equation 9 References 2. Continued Fractions This rather long section gives several crucial tools for solving Pell s equation. Definition.. Let a 0, a, a 2,..., a m be real numbers. Then, a 0 + a + a 2 + a 3 + a a m + a m is called a finite continued fraction and is denoted by [a 0,a,a 2,...,a m ]. If the chain of fractions does not stop, then it is called an infinite continued fraction. Remark.2. From now on, we will use a n as it is defined here. Definitions.3. (a) For n m, [a 0,a,...,a n ] is called nth convergent to [a 0,a,a 2,...,a m ]. (b) Define two sequences of real numbers, (p n ) and ( ), recursively as follows: () p, p 0 a 0, and p n a n p n + p n 2 ; and (2) q 0, q 0, and a n + 2. Remark.4. From now on, we will use p n and as they are defined here. Date: AUGUST 22, 2008.

2 2 SEUNG HYUN YANG Theorem.5. Let [a 0,a,a 2,...,a m ] be a continued fraction. Then, for 0 n m, pn [a 0,a,a 2,...,a n ]. Proof. We proceed by induction. For n, [a 0, a ] a 0 + a a a 0 + a a p 0 + p a q 0 + q p q as desired. Now, suppose the theorem holds for n. [a 0,..., a n+ ] [a 0,..., a n, a n + a n+ ] (a n + a n+ )p n + p n 2 (a n + a n+ ) + 2 (a n+a n + )p n + a n+ p n 2 (a n+ a n + ) + a n+ 2 a n+a n p n + a n+ p n 2 + p n a n+ a n + a n+ 2 + a n+(a n p n + p n 2 ) + p n a n+ (a n + 2 ) + a n+p n + p n a n+ + p n+ + as desired. Remark.6. As a result of this theorem, we will refer to p n as the nth convergent. Theorem.7. Suppose that a 0 is an integer, that a n is a positive integer for each n m, and that a m. Then, p n and are (a) integers and (b) coprime for each n m. Proof. (a) p, q, p 0, and q 0 are integers by definition. a n s are also integers for 0 n m by the given. Since p n and ( n m ) are defined as combinations of multiplication and subtraction of said variables (which are integers), they must be integers as well. (b) We use the following useful lemma: Lemma.8. For n m, following two relations hold: () p n p n ( ) n ; and (2) p n 2 p n 2 ( ) n a n.

3 CONTINUED FRACTIONS AND PELL S EQUATION 3 Proof. () p n p n (a n p n + p n 2 ) p n (a n + 2 ) as desired. (2) p n 2 + a n p n a n p n p n 2 ( )(p n 2 p n 2 )... ( ) n (p 0 q p q 0 ) ( ) n ( ) ( ) n p n 2 p n 2 (a n p n + p n 2 ) 2 (a n + 2 )p n 2 a n p n 2 + p n 2 2 p n 2 2 a n p n 2 a n (p n 2 p n 2 ) () n 2 a n as desired. ( ) n a n To complete the proof of Theorem.7, consider () of Lemma.8. Since there is a linear combination of p n and that is equal to ±, by elementary proposition from number theory, we conclude they are coprime as greatest common divisor between them is at most (equal to). We now move on to use continued fractions to approximate real numbers. Definition.9. Let α be a real number. For n 0,,2,..., define a recursive algorithm as follows: α 0 α, a n α n, and α n a n + α n+. Remark.0. Here, is used as a floor function. That means, by the last equation, α n is positive for all positive n. Observe that α n+ α n - α n. Thus, the left side of the equation must be less than, and therefore α n+ >. Then, by definition, a n+ is a positive integer. Thus, we may conclude that α n,a n for n. Observe also that, given positive m, α [a 0,a,...,a m,α m ]. It is called the mth continued fraction of α. Remark.. Observe that, if α is rational, then the algorithm above is equivalent to Euclidean algorithm, with α n, when reduced to a fraction of coprimes, consisting of nth remainder as numerator and n + th remainder as denominator. That means, the continued fraction of a rational number is finite. On the other hand, if α is irrational, then the continued fraction must be infinite simply because any finite continued fraction is rational (and therefore cannot be equal to an irrational number). Theorem.2. Let α be irrational and p n Then, (.3) α p n (α n+ + ). be a convergent to its continued fraction.

4 4 SEUNG HYUN YANG Proof. Let n be a positive number. Then, α [a 0,a,...,a n,α n+ ]. Then, α p n α n+p n + p n α n+ + p n p n + α n+ p n α n+ p n p n (α n+ + ) p n p n (α n+ + ) ( )(p n p n ) (α n+ + ) ( ) n (α n+ + ) by the Lemma.8. Remark.4. Observe, for each positive n, a n α n by a property of floor function. Also, since q, q 0, and a n (n > 0) are positive integers, the same must be true for (n > 0) by definition. Then, α p n < (α n+ + ) (a n+ + ). + By definition, a n + 2. Since a n and 2 > 0, we conclude that is strictly increasing as n increases. Then, we may conclude that the continued fraction of a number converges to that number by the inequality just given here. In other words, α lim n p n [a 0,a,a 2,... ]. Corollary.5. Let α be a real number with convergent p n. Then, α p n < α p n. Proof. By Theorem.2, α p n α p n (α n+ + ) α n+ + Similarly, α p n α n + 2

5 CONTINUED FRACTIONS AND PELL S EQUATION 5 It now suffices to prove the following inequality: < α n+ + α n + 2 (a n + α n+ ) + 2 a n + α n+ + 2 α n+ a n α n α n+ a n α n+ + + α n+ 2 < α 2 n+ + α n+ α n+ ( a n + 2 ) + < αn+ 2 + α n+ α n+ + < α n+ ( α n+ + ) which is true by Remark.0. < α n+ We are now ready to move onto two extremely beautiful and important approximation theorems involving convergents. Theorem.6. (The Law of Best Approximations) Let α be a real number with convergent pn and n 2. If p,q are integers such that 0 < q and p q p n, then α p n < qα p. Moreover, a reduced fraction p q with q q 2 that satisfies the latter inequality is a convergent. Proof. By Corollary.5, we have already proven the case in which p q is a convergent. Suppose q. Then, p p n. p q p n p p n. α p n < + < 2 because + 3 if n 2 ( is strictly increasing for n and q q 0 ). By the Triangle Inequality, we have α p q p q p n α p n > 2 2 > α p n

6 6 SEUNG HYUN YANG Multiplying both sides by q, we obtain the desired inequality. Now suppose 0 < q <. We may set up following system of two equations with two variables X and Y: (.7) (.8) p n X + p n Y p X + Y q A series of basic manipulations from high school mathematics yields the following unique solution (x,y): x y p qp n p n p n p qp n p n p n By Lemma.8, the denominators reduce to ±: x ±(p qp n ) y ±(p qp n ) Therefore, x and y are nonzero (otherwise p q is a convergent) integers. By the equation.8, since > q, we conclude that x and y have opposite signs. By Lemma.8, α p n alternates signs. That is, α p n and α p n have opposite signs. This implies that α p n and α p n have opposite signs as well. Therefore, x( α p n ) and y( α p n ) have the same sign. Thus, qα p ( x + y)α (p n x + p n y) x( α p n ) + y( α p n ) qα p x( α p n + y( α p n ) qα p > α p n > α p n as desired. In particular, this inequality holds for < q <, which, by Induction Principle, implies only convergents satisfy the said inequality. Theorem.9. (a) Given any two consecutive convergents to a real number α, there is at least one satisfying α p q < 2q 2 (b) Any reduced fraction that satisfies the above inequality is a convergent. Proof. (a) By Lemma.8, given any two consecutive convergents, exactly one is greater than or equal to α, and the other is less than or equal to α. Thus, (.20) p n+ p n + α p n+ + p n α +

7 CONTINUED FRACTIONS AND PELL S EQUATION 7 Now, suppose the said inequality is not true for some consecutive convergents p n and pn+ +. By Lemma.8, p n+ p n p n+ p n + α p n+ + p n α + 2q 2 n+ 2q 2 n q 2 n ( + ) q2 n q 2 n+ 2 Because is strictly increasing for positive n, this may be true only if n 0 and q q 0 a. Thus, by contradiction, (a) is true for all positive n, and we only need to check for the case n 0 (the two consecutive convergents being p0 q 0 and p q ): 0 < p q α a a 0 + α a 0 + α a 0 + [a 0,, a 2, a 3,... ] < + a 2 a 2 a which satisfies the statement of the theorem. (b) Suppose p q satisfies the said inequality. By the law of best approximations, it suffices to show p q is a best approximation to α. Let P Q be such that P Q p α q and Qα P qα p q p < 2q. Then, q Thus, p q qq is a best approximation. pq P q qq p q P Q α p q + P Q α < 2q 2 + 2qQ q + Q 2q 2 Q < q + Q 2q 2q < q + Q q < Q

8 8 SEUNG HYUN YANG We now move on to the discussion of quadratic irrationals. We first present two well-known theorems, Lagrange s and Galois, without proof. Both may be proved with what we have demonstrated so far (plus some basic knowledge of polynomials), and have quite long proofs that would only make this already-too-long paper even longer. Definitions.2. Let α be an irrational number. α is called a quadratic irrational if it is a root of integer polynomial of degree two. The other root β is called a conjugate of α. Definitions.22. Let [a 0,a,a 2,... ] be a continued fraction such that a n a n+l for all sufficiently large n and a fixed positive integer l. Then, it is periodic and l is called a period. Theorem.23. (Lagrange s Theorem) An irrational number is quadratic irrational if and only if its continued fraction is periodic. Definition.24. Let α be a quadratic irrational and β be its conjugate. Then, α is reduced if α > and - < β < 0. Theorem.25. (Galois Theorem) Let α be irrational. Then, α is purely periodic if and only if α is reduced. If α [a 0, a, a 2,..., a l ] and β is its conjugate, then - β [a l,..., a 2, a ]. We finally conclude this section with the following simple theorem: Theorem.26. Let d be a non-square natural number. Then, there exist integers a, a 2,..., a n such that [ d,a, a 2,..., a 2, a, 2 d ] is the continued fraction of d. Also, a, a 2,..., a 2, a is a palindrome. Proof. Observe that - < α d - d < 0 and < β d + d. (x α)(x β) x 2 (α + β)x + αβ x d x + d 2 d which is an integer polynomial of degree two. Thus, α and β are quadratic irrationals and conjugates of each other. More, β is reduced. By Galois Theorem, β is purely periodic. β d + d [2 d, a, a 2,..., a l ] On the other hand, observe: d d [2 d, a, a 2,..., a l, 2 d ] d [ d, a, a 2,..., a l, 2 d ] d + d [a l,..., a 2, a, 2 d ] by Galois Theorem. That means, d [ d, d d ] [ d, a l,..., a 2, a, 2 d ] Thus, we have obtained the desired palindrome.

9 CONTINUED FRACTIONS AND PELL S EQUATION 9 2. Solution to Pell s Equation Definition 2.. Let d be a natural number. Then, Pell s equation is X 2 dy 2. Remarks 2.2. The case in which d is a perfect square is trivial. Let d m 2. X 2 dy 2 (X my )(X + my ) X my ± X + my ± Solving this linear system of equations yields the only solutions: (±,0). Note that these are solutions regardless of d. Observe that if (x,y) is a solution, then ( x, y) and (±x, y) are also solutions. Thus, it suffices to consider only positive solutions. For convenience, if (x,y) is a solution, then we call x + y d a solution as well. Theorem 2.3. (a) Let d not be a perfect square and α d. Note that α n is as defined in Definition.9. For nonnegative n, there exist integers Q n and P n such that α n P n+ d Q n and that d Pn 2 0 (mod Q n ). (b) For each n 2, we have: (2.4) p 2 n dq 2 n ( ) n Q n Proof. (a) We proceed by induction. For n 0, α 0 d, Q 0, and P 0 0. For n, α d d d d d d 2 Now assume the statement for n. α n+ Q d d 2 and P d α n a n P n + d Q n a n Q n P n + d a n Q n Q n(p n a n Q n d) (P n a n Q n ) 2 d : P n+ + d Q n+ P n+ a n Q n P n, which is integral, and Q n+ d (P n a n Q n ) 2 Q n d P 2 n Q n + 2a n P n a 2 nq n,

10 0 SEUNG HYUN YANG which is integral because d P 2 n 0 (mod Q n ) by assumption. Finally, as desired. (b) Q n d (P n a n Q n ) 2 Q n+ d P n+ Q n+ d P 2 n+ 0 (mod Q n ) d α n p n + p n 2 α n + 2 (P n + d)p n + p n 2 Q n (P n + d) + 2 Q n d((p n + d) + 2 Q n ) (P n + d)p n + p n 2 Q n d(p n + Q n 2 + d ) p n d + Pn p n + p n 2 Q n (P n + Q n 2 ) d + d p n d + Pn p n + p n 2 Q n p n P n + 2 Q n and d P n p n + p n 2 Q n Multiply the first by p n and the second by, and obtain: p 2 n dq 2 n Q n (p n 2 p n 2 ) p 2 n dq 2 n ( ) n Q n by Lemma.8. Definition 2.5. Given Pell s equation with some d, let (x,y) be the positive solution with the minimum x. Such solution is called the minimal solution. We finally give the complete solution to Pell s equation: Theorem 2.6. Let variables be as defined in the previous theorem. Let l be the minimal period of the continued fraction of d. (a) The minimal solution to Pell s equation is: { (p l, q l ) if l is even (x,y ) (p 2l, q 2l ) if l is odd (b) All the solutions (x,y) to Pell s equation are up to sign given by the powers of the minimal solution (x,y ): x + y d ±(x + y d) ±n (n 0,,2,... )

11 CONTINUED FRACTIONS AND PELL S EQUATION Proof. (a) We first establish that all the solutions to Pell s equation X 2 dy 2 must consist of convergents to d. Let (x,y) be a solution. Observe first that: x d y d x y y dy 2 x 2 y(y d + x) y 2 ( d + x y ) < (since d > and x > y) 2y2 By Theorem.9, we conclude (x,y) is a convergent to d. We now use (2.4) to show that: () Q n for every positive integer n; and (2) Q n if and only if n is a multiple of l. Thus, with regard to (2.4), the minimal solution is obtained by taking the smallest positive odd multiple of l minus, as (a) states. To show (), observe that α α n+ if and only if n is a multiple of l because l is minimal. Let Q n. Then, by Theorem 2.3, α n P n + d. Being quadratic irrational, by Lagrange s Theorem, α n has purely periodic continued fraction expansion. By Galois Theorem, α n is reduced: < β n P n d < 0 d < P n < d P n d Thus, α d + d. This implies that α n+ α, and n is a multiple of l. On the other hand, if n is a multiple of l, n kl, then: P kl + d Q kl α kl [0, a, a 2,..., a l ] d d Q kl Q kl, as desired. For (2), suppose Q n. By Theorem 2.3, α n P n d. α n is quadratic irrational, and thus is reduced by Galois Theorem: < P n + d < 0 and < P n d d < P n < d + d < 2 This is contradiction. Thus, Q n for all n, as desired. (b) We observed before that, given a solution (x,y), signs of x and/or y may be changed without contradicting the statement x 2 dy 2. We then concluded that it sufficed to find positive solutions. With regard to this, observe that ± x + y d ±(x y d) x 2 dy 2 ±x y d (x 2 dy 2 ) Thus, we may change the signs of x and y only by taking powers and multiplying by -. Thus, it now suffices to show this theorem for only positive solutions.

12 2 SEUNG HYUN YANG Let ɛ : x + y d and (x,y) be a positive solution to Pell s equation. Then, there exists n such that ɛ n x + y d < ɛ n+. Now, let X + Y d ɛ n (x + y d). Observe that, due to the definition of ɛ, X + Y d. It suffices to show that it is equal to. We do this by contradiction; suppose ɛ > X + Y d > (the first inequality is due to the definition of epsilon also). By rules of elementary algebra, we get the following conjugation to this quadratic irrational: X Y d ɛ n (x y d). Then, (X + Y d)(x Y d) X 2 dy 2 ɛ n ɛ n (x y d)(x + y d) x 2 dy 2. We then have: 0 < ɛ n < (X + Y d) X Y d < { 2X (X + Y d) + (X Y d) > + ɛ > 0 2Y d (X + Y d) (X Y d) > 0 Thus, (X, Y ) is a positive solution with X + Y d < x + y d. That is, X < x. This is a contradiction to the minimality of (x, y ), and thus our assumption (X + Y d > ) is false, as desired. Remarks 2.7. In fact, (a) alone suffices to give the complete set of solutions to Pell s equation. The set of (positive) solutions deduced from (a) would be: { (p kl, q kl ) if l is even (x k, y k ) (p 2kl, q 2kl ) if l is odd (b) shows that the complete set of solutions to Pell s equation is infinite cyclic group generated by the minimal solution. References [] Jörn Steuding. Diophantine Analysis. Chapman and Hall/CRC [2] Titu Andreescu and Dorin Andrica. An Introduction to Diophantine Equations. Gil Publishing House

Continued Fractions and the Euclidean Algorithm

Continued 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 information

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS

CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS CONTINUED FRACTIONS, PELL S EQUATION, AND TRANSCENDENTAL NUMBERS JEREMY BOOHER Continued fractions usually get short-changed at PROMYS, but they are interesting in their own right and useful in other areas

More information

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

More information

HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

HOMEWORK 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 information

CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 Solutions CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

More information

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen CONTINUED FRACTIONS AND FACTORING Niels Lauritzen ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK EMAIL: niels@imf.au.dk URL: http://home.imf.au.dk/niels/ Contents

More information

k, then n = p2α 1 1 pα k

k, 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 information

Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.

Elementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook. Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole

More information

GREATEST COMMON DIVISOR

GREATEST 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 information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

Further linear algebra. Chapter I. Integers.

Further linear algebra. Chapter I. Integers. Further linear algebra. Chapter I. Integers. Andrei Yafaev Number theory is the theory of Z = {0, ±1, ±2,...}. 1 Euclid s algorithm, Bézout s identity and the greatest common divisor. We say that a Z divides

More information

Continued fractions and good approximations.

Continued 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 information

Chapter 3: Elementary Number Theory and Methods of Proof. January 31, 2010

Chapter 3: Elementary Number Theory and Methods of Proof. January 31, 2010 Chapter 3: Elementary Number Theory and Methods of Proof January 31, 2010 3.4 - Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Quotient-Remainder Theorem Given

More information

1.3 Induction and Other Proof Techniques

1.3 Induction and Other Proof Techniques 4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

More information

Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

More information

On continued fractions of the square root of prime numbers

On continued fractions of the square root of prime numbers On continued fractions of the square root of prime numbers Alexandra Ioana Gliga March 17, 2006 Nota Bene: Conjecture 5.2 of the numerical results at the end of this paper was not correctly derived from

More information

SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me

SYSTEMS OF PYTHAGOREAN TRIPLES. Acknowledgements. I would like to thank Professor Laura Schueller for advising and guiding me SYSTEMS OF PYTHAGOREAN TRIPLES CHRISTOPHER TOBIN-CAMPBELL Abstract. This paper explores systems of Pythagorean triples. It describes the generating formulas for primitive Pythagorean triples, determines

More information

MATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:

MATH 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 information

Appendix F: Mathematical Induction

Appendix F: Mathematical Induction Appendix F: Mathematical Induction Introduction In this appendix, you will study a form of mathematical proof called mathematical induction. To see the logical need for mathematical induction, take another

More information

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I. GROUPS: BASIC DEFINITIONS AND EXAMPLES I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

More information

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b. Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

More information

PYTHAGOREAN TRIPLES KEITH CONRAD

PYTHAGOREAN 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

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom. Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

More information

Homework until Test #2

Homework until Test #2 MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

More information

4.3 Limit of a Sequence: Theorems

4.3 Limit of a Sequence: Theorems 4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,

More information

2 The Euclidean algorithm

2 The Euclidean algorithm 2 The Euclidean algorithm Do you understand the number 5? 6? 7? At some point our level of comfort with individual numbers goes down as the numbers get large For some it may be at 43, for others, 4 In

More information

Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB

Problem Set 5. AABB = 11k = (10 + 1)k = (10 + 1)XY Z = XY Z0 + XY Z XYZ0 + XYZ AABB Problem Set 5 1. (a) Four-digit 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 information

Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

More information

Introduction. Appendix D Mathematical Induction D1

Introduction. Appendix D Mathematical Induction D1 Appendix D Mathematical Induction D D Mathematical Induction Use mathematical induction to prove a formula. Find a sum of powers of integers. Find a formula for a finite sum. Use finite differences to

More information

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013

Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of

More information

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

More information

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins

MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins MA2C03 Mathematics School of Mathematics, Trinity College Hilary Term 2016 Lecture 59 (April 1, 2016) David R. Wilkins The RSA encryption scheme works as follows. In order to establish the necessary public

More information

ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

ELEMENTARY NUMBER THEORY AND METHODS OF PROOF CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Copyright Cengage Learning. All rights reserved.

More information

Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2)

Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2) Solutions to Homework Set 3 (Solutions to Homework Problems from Chapter 2) Problems from 21 211 Prove that a b (mod n) if and only if a and b leave the same remainder when divided by n Proof Suppose a

More information

The Topsy-Turvy World of Continued Fractions [online]

The Topsy-Turvy World of Continued Fractions [online] Chapter 47 The Topsy-Turvy World of Continued Fractions [online] The other night, from cares exempt, I slept and what d you think I dreamt? I dreamt that somehow I had come, To dwell in Topsy-Turveydom!

More information

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

More information

It is time to prove some theorems. There are various strategies for doing

It is time to prove some theorems. There are various strategies for doing CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

More information

A Curious Case of a Continued Fraction

A Curious Case of a Continued Fraction A Curious Case of a Continued Fraction Abigail Faron Advisor: Dr. David Garth May 30, 204 Introduction The primary purpose of this paper is to combine the studies of continued fractions, the Fibonacci

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. 1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

More information

12 Greatest Common Divisors. The Euclidean Algorithm

12 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 information

MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

More information

Prime Numbers. Chapter Primes and Composites

Prime Numbers. Chapter Primes and Composites Chapter 2 Prime Numbers The term factoring or factorization refers to the process of expressing an integer as the product of two or more integers in a nontrivial way, e.g., 42 = 6 7. Prime numbers are

More information

Contents. 6 Continued Fractions and Diophantine Equations. 6.1 Linear Diophantine Equations

Contents. 6 Continued Fractions and Diophantine Equations. 6.1 Linear Diophantine Equations Number Theory (part 6): Continued Fractions and Diophantine Equations (by Evan Dummit, 04, v 00) Contents 6 Continued Fractions and Diophantine Equations 6 Linear Diophantine Equations 6 The Frobenius

More information

MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

More information

7. Some irreducible polynomials

7. Some irreducible polynomials 7. Some irreducible polynomials 7.1 Irreducibles over a finite field 7.2 Worked examples Linear factors x α of a polynomial P (x) with coefficients in a field k correspond precisely to roots α k [1] of

More information

Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D.

Stanford University Educational Program for Gifted Youth (EPGY) Number Theory. Dana Paquin, Ph.D. Stanford University Educational Program for Gifted Youth (EPGY) Dana Paquin, Ph.D. paquin@math.stanford.edu Summer 2010 Note: These lecture notes are adapted from the following sources: 1. Ivan Niven,

More information

This section demonstrates some different techniques of proving some general statements.

This section demonstrates some different techniques of proving some general statements. Section 4. Number Theory 4.. Introduction This section demonstrates some different techniques of proving some general statements. Examples: Prove that the sum of any two odd numbers is even. Firstly you

More information

On the largest prime factor of x 2 1

On the largest prime factor of x 2 1 On the largest prime factor of x 2 1 Florian Luca and Filip Najman Abstract In this paper, we find all integers x such that x 2 1 has only prime factors smaller than 100. This gives some interesting numerical

More information

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

Elementary Number Theory We begin with a bit of elementary number theory, which is concerned CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

More information

Settling a Question about Pythagorean Triples

Settling a Question about Pythagorean Triples Settling a Question about Pythagorean Triples TOM VERHOEFF Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-Mail address:

More information

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

More information

The last three chapters introduced three major proof techniques: direct,

The last three chapters introduced three major proof techniques: direct, CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

More information

STUDENT S SOLUTIONS MANUAL ELEMENTARY NUMBER THEORY. Bart Goddard. Kenneth H. Rosen AND ITS APPLICATIONS FIFTH EDITION. to accompany.

STUDENT S SOLUTIONS MANUAL ELEMENTARY NUMBER THEORY. Bart Goddard. Kenneth H. Rosen AND ITS APPLICATIONS FIFTH EDITION. to accompany. STUDENT S SOLUTIONS MANUAL to accompany ELEMENTARY NUMBER THEORY AND ITS APPLICATIONS FIFTH EDITION Bart Goddard Kenneth H. Rosen AT&T Labs Reproduced by Pearson Addison-Wesley from electronic files supplied

More information

We now explore a third method of proof: proof by contradiction.

We now explore a third method of proof: proof by contradiction. CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

More information

Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

More information

Lectures on Number Theory. Lars-Åke Lindahl

Lectures on Number Theory. Lars-Åke Lindahl Lectures on Number Theory Lars-Åke Lindahl 2002 Contents 1 Divisibility 1 2 Prime Numbers 7 3 The Linear Diophantine Equation ax+by=c 12 4 Congruences 15 5 Linear Congruences 19 6 The Chinese Remainder

More information

Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

More information

a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

a 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 information

it is easy to see that α = a

it is easy to see that α = a 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore

More information

Continued Fractions. Darren C. Collins

Continued 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 information

H/wk 13, Solutions to selected problems

H/wk 13, Solutions to selected problems H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.

More information

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions

March 29, 2011. 171S4.4 Theorems about Zeros of Polynomial Functions MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 4: Polynomial and Rational Functions 4.1 Polynomial Functions and Models 4.2 Graphing Polynomial Functions 4.3 Polynomial

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: 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 information

Zeros of a Polynomial Function

Zeros of a Polynomial Function Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

More information

8 Divisibility and prime numbers

8 Divisibility and prime numbers 8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

More information

x if x 0, x if x < 0.

x 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 information

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain

1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain Notes on real-closed fields These notes develop the algebraic background needed to understand the model theory of real-closed fields. To understand these notes, a standard graduate course in algebra is

More information

ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

ELEMENTARY NUMBER THEORY AND METHODS OF PROOF CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF Copyright Cengage Learning. All rights reserved. SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem

More information

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I. A brief introduction to proofs, sets, and functions MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

More information

0.8 Rational Expressions and Equations

0.8 Rational Expressions and Equations 96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

More information

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

More information

Every Positive Integer is the Sum of Four Squares! (and other exciting problems)

Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer

More information

Mathematical Induction

Mathematical Induction Mathematical Induction Victor Adamchik Fall of 2005 Lecture 1 (out of three) Plan 1. The Principle of Mathematical Induction 2. Induction Examples The Principle of Mathematical Induction Suppose we have

More information

PYTHAGOREAN TRIPLES PETE L. CLARK

PYTHAGOREAN TRIPLES PETE L. CLARK PYTHAGOREAN TRIPLES PETE L. CLARK 1. Parameterization of Pythagorean Triples 1.1. Introduction to Pythagorean triples. By a Pythagorean triple we mean an ordered triple (x, y, z) Z 3 such that x + y =

More information

SUM OF TWO SQUARES JAHNAVI BHASKAR

SUM OF TWO SQUARES JAHNAVI BHASKAR SUM OF TWO SQUARES JAHNAVI BHASKAR Abstract. I will investigate which numbers can be written as the sum of two squares and in how many ways, providing enough basic number theory so even the unacquainted

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5. Number Theory. 1. Integers and Division. Discussion CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

Answer Key for California State Standards: Algebra I

Answer Key for California State Standards: Algebra I Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

More information

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

More information

3. Mathematical Induction

3. Mathematical Induction 3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

More information

Congruent Number Problem

Congruent Number Problem University of Waterloo October 28th, 2015 Number Theory Number theory, can be described as the mathematics of discovering and explaining patterns in numbers. There is nothing in the world which pleases

More information

Mathematical Induction

Mathematical Induction Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

More information

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0

THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 THE NUMBER OF REPRESENTATIONS OF n OF THE FORM n = x 2 2 y, x > 0, y 0 RICHARD J. MATHAR Abstract. We count solutions to the Ramanujan-Nagell equation 2 y +n = x 2 for fixed positive n. The computational

More information

Chapter Two. Number Theory

Chapter Two. Number Theory Chapter Two Number Theory 2.1 INTRODUCTION Number theory is that area of mathematics dealing with the properties of the integers under the ordinary operations of addition, subtraction, multiplication and

More information

Congruences. Robert Friedman

Congruences. Robert Friedman Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition

More information

Some Notes on Taylor Polynomials and Taylor Series

Some 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 information

Math 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 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 information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

Handout NUMBER THEORY

Handout NUMBER THEORY Handout of NUMBER THEORY by Kus Prihantoso Krisnawan MATHEMATICS DEPARTMENT FACULTY OF MATHEMATICS AND NATURAL SCIENCES YOGYAKARTA STATE UNIVERSITY 2012 Contents Contents i 1 Some Preliminary Considerations

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

More information

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers Amnon Yekutieli Department of Mathematics Ben Gurion University email: amyekut@math.bgu.ac.il Notes available at http://www.math.bgu.ac.il/~amyekut/lectures

More information

Notes on Factoring. MA 206 Kurt Bryan

Notes on Factoring. MA 206 Kurt Bryan The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

More information

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

More information

Algebra 2. Rings and fields. Finite fields. A.M. Cohen, H. Cuypers, H. Sterk. Algebra Interactive

Algebra 2. Rings and fields. Finite fields. A.M. Cohen, H. Cuypers, H. Sterk. Algebra Interactive 2 Rings and fields A.M. Cohen, H. Cuypers, H. Sterk A.M. Cohen, H. Cuypers, H. Sterk 2 September 25, 2006 1 / 20 For p a prime number and f an irreducible polynomial of degree n in (Z/pZ)[X ], the quotient

More information