CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen

Size: px
Start display at page:

Download "CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen"

Transcription

1 CONTINUED FRACTIONS AND FACTORING Niels Lauritzen

2 ii NIELS LAURITZEN DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF AARHUS, DENMARK URL:

3 Contents 1 Factoring using continued fractions The Fermat-Kraitchik method Continued fractions The game that might never stop Rational numbers Basic theory of continued fractions Eulers rule and corollaries Continued fraction for a real number Quadratic irrationalities Purely periodic continued fractions The continued fraction for Æ A few words on Pells equation Exercises Inclass Homework iii

4 iv CONTENTS

5 Chapter 1 Factoring using continued fractions The statement that every integer can be written as a product of prime numbers is a typical mathematical statement with a simple proof. Things become much more complicated when you (inspired by Gauss) ask for a good algorithm for factoring a given integer Æ. In a non-trivial factorization Æ one of the factors and must be Æ.IfÆ is even ¾ divides and we have found a factor. If Æ is odd we may find a factor of Æ, by starting with and try dividing with odd numbers up to Æ. This procedure is called trial division. The number of steps in trial division is proportional to the smallest prime factor. This is extremely slow. If you want to factor a ¼¼ digit number, which is the product of two ¼ digit prime numbers, you must carry out approximately ¼ ¼ steps of trial division. If every step takes ¼ ¼ seconds, you will have to wait for ¼ ¼ seconds (or approximately ¼ ¾ years). There are better algorithms The Fermat-Kraitchik method Currently the most effective algorithms for factoring difficult integers originates in the historic fact that if an integer Æ can be written as the difference Ü ¾ Ý ¾ between two squares, we have the factorization Æ Ü ¾ Ý ¾ Ü Ýµ Ü Ýµ On the other hand if an odd number Æ ÙÚ is composite, then Ù Ú ¾ Ù Ú ¾ Æ ¾ ¾ This method of factoring goes back to Fermat. Suppose we wish to factor Æ. Fermat s method would start with the function Ë Üµ Ü ¾ Æ 1

6 2 CHAPTER 1. FACTORING USING CONTINUED FRACTIONS and search for Ü, such that Ë Üµ Ý ¾ is square. Usually one runs through Ü Æ Ü Æ, where Æ denotes the integral part of Æ. Putting Æ ¾, one would find Ë µ ¼Ë ¼µ ¾. This means that ¾ ¼ µ ¼ µ. Of course using this method on a composite number (like ¾ ¼¼¼ ) works just as terribly as trial division. There is a beautiful variation of Fermat s method (due to M. Kraitchik ( )) using congruences. The insight is that it usually suffices that Æ divides Ü ¾ Ý ¾ to find a factor of Æ. This means that Æ Ü ¾ Ý ¾ Ü Ýµ Ü Ýµ Now if Æ does not divide any of Ü Ý and Ü Ý, then we may conclude that Ü Ý Æµ and use the Euclidean algorithm to find ÆÜ Ýµ, which is a non-trivial factor of Æ. So one should look for solutions Ü, Ý such that Ü ¾ Ý ¾ ÑÓ Æ µ Ü Ý ÑÓ Æ µ Suppose we have collected Ü Ü Ò, such that Ü ¾ ÑÓ Æ µü ¾ Ò Ò ÑÓ Æ µ for some integers Ò. If a subset Ö of ¾ Ò satisifes that Ö is a square, then Ü Ü Ö µ ¾ Ö ÑÓ Æ µ and we have our congruence Ü ¾ Ý ¾ ÑÓ Æ µ. This congruence may or may not satisfy Ü Ý ÑÓ Æ µ. To tell if a number Ò is square we factor it Ò Ò ÒÖ Ö using some predefined factor basis È Ö of (small) prime numbers. Now Ò is a square if and only if all the exponents Ò Ò Ö are even. Exercise Suppose that the prime factorizations of Ò over the factor basis È (assume all a s factor completely using primes from È )are Ñ ¾ Ñ ¾. Ò Ñ Ò Ñ Ö Ö Ñ ¾Ö Ö ÑÒÖ Ö Use linear algebra over ¾ to find a subset Ö of ¾Ò such that Ö is a square.

7 3 Let us apply this to the numbers we get from the function Ë Üµ Ü ¾ Æ. Notice that Ü ¾ Ë Üµ ÑÓ Æ µ. We wish to find values Ü Ü Ò, such that the product of a suitable subset of the numbers Ë Ü µë Ü Ò µ is a square. For Æ ¾¼(this example is from [1]) we illustrate this in the table below Ü Ü ¾ Æ Factorization Marked ¾ ¾ ¾ ¾ ¾ ¾ The above table shows that Ë µë µë µë µ ¼ ¼ ¾ ¾ ¾ µ ¾ is a square. Putting Ù ¾ ¾ ¾, we get Ù ¾ ¼¼¼ ¾ ¾ ÑÓ ¾¼µ. Now we know that Ù ¾ Ú ¾ ÑÓ ¾¼µ where Ú ¼¾ ÑÓ ¾¼µ. Using the Euclidean algorithm one finds the greatest common divisor of Ù Ú ¼ and ¾¼, which is. We have found the factorization ¾¼. Using the original method of Fermat we would have to wait until Ü ¼ before having a subset whose product is a square. The heavy part of the algorithm is factoring Ë Üµ Ü ¾ Æ. Around 1982 Pommerance discovered a nice trick to avoid this. The observation is that a prime power Ö divides Ë Üµ if and only if it divides Ë Ü Ö µ, where ¾. So if one can locate a number Ü such that Ö Ë Üµ, then we know in advance that Ö Ë Ü Ö µë Ü ¾ Ö µ. This is a socalled sieving procedure (like the sieve of Eratosthenes eliminating multiples of prime numbers). It leads to the factorization algorithm called the quadratic sieve. In [1] you can find a nice description of this and other sieving methods for factoring. These are currently the most effective factoring the challenges issued by RSA. In fact the RSA challenge with digits was factored using sieving. We will however describe the champion of factoring preceeding the quadratic sieve, the continued fraction algorithm. This is in order to get involved in some fantastic 19th century mathematics and show how an idea from the heart of mathematics can be applied in easing factoring. The problem with Fermats method is that Ë Üµ Ü ¾ Æ grows too rapidly. It takes longer and longer to factor Ë Üµ. One can instead use convergents Ò Ø Ò in the continued fraction expansion of Æ (or Æ, where ¾ Æ). Then one tries the above method for factoring successively using the numbers Ü Ò Ò Ý Ò ¾ Ò Æؾ Ò

8 4 CHAPTER 1. FACTORING USING CONTINUED FRACTIONS Clearly Ü ¾ Ò Ý Ò ÑÓ Æ µ. From the theory of continued fractions one gets the inequality Ý Ò ¾ Æ Exercise Prove this inequality after having read about continued fractions in the next chapter.

9 Chapter 2 Continued fractions 2.1 The game that might never stop Let Ü denote the largest integer Ü, where Ü ¾ Ê is a real number. For a given real number we wish to decide if is rational. Clearly this is true if. If not, we may write Now put.if, was a rational number. If not we put ¾ and write ¾ We continue this and cook up new real numbers and stop if Ò Ò for some Ò. This game is called the continued fraction algorithm for a real number. The game never stops if is an irrational number. Exercise Prove this! Example The first steps of the continued fraction algorithm for leads 5

10 6 CHAPTER 2. CONTINUED FRACTIONS to the following continued fraction where is an irrational number. ¾¾ Example The continued fraction algorithm for ¾ can be carried out by algebraic computations. Here is how. First we rewrite a bit ¾ ¾ ¾ Now we have an expression for ¾ that bites its own tail. Let us insert it into itself: ¾ ¾ ¾ ¾ We can repeat this to get the continued fraction ¾ ¾ ¾ ¾ In this way we have proved that ¾ is irrational or have we? Rational numbers The continued fraction algorithm for rational numbers turns out to be the classical Euclidean algorithm. This is quite easy to see. Consider a fraction, where ¼. Then Õ Ö and Õ. Therefore Õ Ö Õ Ö

11 2.2. BASIC THEORY OF CONTINUED FRACTIONS 7 and the continued fraction algorithm continues with the fraction Ö. Ultimately this process will stop. Example Consider the fraction ¼ ¼¾. Division with remainder gives ¼ ¼¾. This implies that ¼ ¼¾ ¼¾ Again ¼¾ ¾, Therefore ¼¾ Continue with ¾ ¾¾: ¾ ¾ ¾ ¾¾ ¾ ¼¾ 2.2 Basic theory of continued fractions ¾¾ A continued fraction is formally a sequence of integers ¼ Ò, where ¾ and ¼ for ¼. There is a one to one correspondence between continued fractions and real numbers. This is displayed in the notation ¼ ( ) ¾ to be understood as the following sequence of numbers ¼ ¼ ¼ ¾ ¼ ¾ Does this make sense? Does this sequence converge (to a real number)? We need to compute a bit more to answer this question.

12 8 CHAPTER 2. CONTINUED FRACTIONS 2.3 Eulers rule and corollaries The above sequence of (honest) fractions are called convergents for the continued fraction in ( ). What are the fractions? Before we compute them let us make a subtle observation.denote the numerator of the Ò-th convergent of a continued fraction Ü Ü ¾ Ü by Ü Ü ¾ Ü Ò. Then the Ò-th convergent is Therefore Using this we get Ü Ü ¾ Ü Ò Ü ¾ Ü Ò Ü Ü Ü Ò Ü ¾ Ü Ò Ü Ü ¾ Ü Ò Ü Ü ¾ Ü Ò Ü Ü Ò Ü Ü Ü Ü ¾ Ü Ü ¾ Ü Ü ¾ Ü Ü Ü ¾ Ü Ü Ü Ü Ü ¾ Ü Ü Ü Ü ¾ Ü Ü Ü Ü Ü ¾ Ü Ü Ü ¾ Ü Ü ¾ Ü Ü Ü Ü Ü ¾ Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü Ü ¾ Ü Ü Ü ¾ Ü Ü Ü Notice that Ü Ü ¾ Ü Ü Ü Ü Ü Ü Ü ¾ Ü. This is a very pleasant surprise and it holds in general! In fact we have the following result. Proposition (Eulers rule) Ü Ü ¾ Ü Ò is a sum of terms constructed from Ü Ü ¾ Ü Ò by first deleting ¼ consecutive variables, then deleting ¾ consecutive variables, then and so on. Proof. Follows by induction using Ü Ü ¾ Ü Ò Ü Ü ¾ Ü Ò Ü Ü Ò Corollary Ü Ü ¾ Ü Ò Ü Ò Ü Ò Ü Corollary ¼ Ò Ò ¼ Ò ¼ Ò ¾ Exercise Write the continued fraction for a fraction ÙÚ, where Ù Ú ¼ as ¼ Ò ¾ Ò, where Ù Úµ. Conclude that if the Euclidean algorithm requires precisely Ò steps for Ù and

13 2.3. EULERS RULE AND COROLLARIES 9 Ú, then Ù Ò ¾ and Ú Ò, where Ò denotes the Ò-th Fibonacci number. Here is the list of the first few Fibonacci numbers ¾ ¾ How many steps of the Euclidean algorithm do Ù Ò ¾ and Ú Ò take? If we denote the numerator and denominator of the Ò-th convergent Ò and Ø Ò respectively, then we have the inductive formula Ò Ò Ò Ò ¾ Ø Ò Ò Ø Ò Ø Ò ¾ where we put ¾ ¼ Ø ¾ Ø ¼. Remark The sequence of denominators Ø Ò µ is a strictly increasing sequence of natural numbers. Example Here is the beginning of the continued fraction for ¾ Ø Example Here are the first convergents in the continued fraction for (cf. Example 2.1.2) Ø We recognize in particular ¾¾ as the archimedian approximation to. Less known is, which is a much better approximation. In fact whereas ¾. ¾ Proposition Ò Ø Ò Ò Ø Ò µ Ò The numerator Ò and denominator Ø Ò in a convergent Ò Ø Ò are relatively prime integers.

14 10 CHAPTER 2. CONTINUED FRACTIONS Proof. Write Ò Ò Ò Ò and Ø Ò Ò Ø Ò Ø Ò. Then Ò Ø Ò Ò Ø Ò Ò Ò Ø Ò Ø Ò µ Ò Ò Ò µø Ò Ò Ø Ò Ò Ø Ò µ and the result follows by induction. Notice that ¾Ø Ø ¾. Corollary We have the following inequalities ¼ Ø ¼ ¼ Ø ¼ Ø ¾ Ø ¾ Ø Ø ¾ Ø ¾ Ø Ø The even convergents form an increasing sequence bounded above by Ø and the odd convergents form a decreasing sequence bounded below by ¼ Ø ¼. By elementary real analysis both sequences ¾Ò Ø ¾Ò µ and ¾Ò Ø ¾Ò µ have a limit. In fact they converge to the same number. This is a result of the following computation. Lemma Proof. Ò Ø Ò Ò Ø Ò Ò Ø Ò Ø ¾ Ò Ò ÒØ Ò Ø Ò Ò Ø Ò Ø Ò Ø Ò Ø ¾ Ò by Proposition and since Ø Ò is an increasing sequence of natural numbers. 2.4 Continued fraction for a real number We pose a very relevant question. When we do the continued fraction algorithm for a real number, we get a continued fraction. What does this continued fraction have to do with? The answer is that the convergents (surprise!) converge to. Here is how to prove this. Consider the continued fraction algorithm for after Ò steps ¼ ¾ Ò Ò

15 2.5. QUADRATIC IRRATIONALITIES 11 This means that ¼ Ò Ò Ò Ò. Similarly to Lemma , we get the following result showing that the convergents really do converge to. Proposition Proof. We may write Ò Ø Ò Ø ¾ Ò Ò Ò Ò Ò Ø Ò Ø Ò where Ò ¼. Using this, a small computation shows what we want. 2.5 Quadratic irrationalities A quadratic irrationality «is a non rational real root in a quadratic equation where ¾. Ü ¾ Ü ¼ ( ) Definition If «is a quadratic irrationality, which is a root of ( ), then we let «¼ denote the other root of ( ). The other root is called the (algebraic) conjugate of «. Proposition Let «be a quadratic irrationality and Õ ¾ an integer. Then i) «is a quadratic irrationality. ii) «Õ is a quadratic irrationality. iii) «Õµ ¼ «¼ Õ and «µ ¼ «¼. Proof. Exercise. The above proposition shows that if «is a quadratic irrationality and «Õ «where Õ «, then «is also a quadratic irrationality. In other words: all the steps in the continued fraction algorithm produce quadratic irrationalities when starting out with one. If «is a quadratic irrationality, then «È É

16 12 CHAPTER 2. CONTINUED FRACTIONS for È É ¾, where ¼. To carry out the first step in the continued fraction algorithm, first observe that Exercise Prove this! È «É So if we have a good algorithm (we do) for finding the floor of the square root of an integer we are all set. Let us analyze the inversion step in the continued fraction algorithm. Here È È É È ¾ Notice that É È ¾, since È ¾ and É ¾, where «¾ «¼ and ¾. These observations give a very explicit integer algorithm for computing the continued fraction of a quadratic irrationality. But we are not satisfied! We will dive into the mind blowing theory of continued fractions in the 19th century proving a beautiful result of Galois. É Purely periodic continued fractions Definition A quadratic irrationality «is called reduced if i) «ii) «¼ ¼ Example ¾ is not reduced since ¾µ ¼ ¾. But ¾ is reduced as ¾µ ¼ ¾. In general Õ ¼ Æ is reduced where Õ ¼ Æ. A continued fraction of the form ¼ Ò ¼ is called purely periodic. Example Consider the real number ³ given by the simplest purely periodic continued fraction ³

17 2.5. QUADRATIC IRRATIONALITIES 13 Then ³ ³ Therefore ³¾ ³ ¼and ³ ¾ Some will recognize the part as the golden ratio. The golden ratio is a reduced quadratic irrational. Lemma If «is a reduced quadratic irrationality and «Õ ¼ «where Õ ¼ «. Then «is a reduced quadratic irrationality. Proof. Exercise. Theorem (Galois) ¾ Ê has a purely periodic continued fraction if and only if is a reduced quadratic irrational. Proof. Suppose that ¾ Ê has a purely periodic continued fraction. This means that ¼ ¾ Ò for some Ò. Then Ò Ò Ø Ò Ø Ò and must be a quadratic irrationality, since Ø Ò ¾ Ø Ò Ò µ Ò ¼ But why is it reduced? This is tricky. In fact one has to pull out a genuine trick to solve this. Consider the number given by reversing the period of : Ò Ò Ò ¾ ¼

18 14 CHAPTER 2. CONTINUED FRACTIONS Then ¼ Ò ¼ Ò ¼ Ò Ø Ò Ò Ø Ò Ò ¼ Ò Ò ¼ Ò and Ò ¾ Ø Ò Ò µ Ø Ò ¼ This shows that ¼, so that ¼ ¼. We have proved that a purely periodic continued fraction describes a reduced quadratic irrational. Let us prove the other way. Suppose that is a reduced quadratic irrational. Then È É Ò È ¼ É We may conclude first that É ¼ (consider important boundedness conditions ¼ ), then È ¼ and the i) È ii) ɾ on È and É by using that is reduced. Let us run through one step of the continued fraction algorithm. First let Õ. Then Õ È ÕÉ É Next step is to compute the reduced (recall Lemma 2.5.7) quadratic irrationality È ÕÉ É É È ÕÉ É È Õɵ µ È Õɵ ¾ So putting È ÕÉ È and É È ¾ µé we get È É Now we continue the algorithm with. Since there are only finitely many possibilities for È and É (there are only finitely many pairs È Éµ of natural numbers satisfying È É ¾ ), we will eventually run into a repetition Ñ Ò for ÑÒ. We will prove that this implies Ñ Ò. This

19 2.6. THE CONTINUED FRACTION FOR Æ 15 leads to the purely periodic continued fraction ¼ Ò Ñ ¼. In the Ò-th step of the continued fraction algorithm we have This implies that Ò Ò Ò ¼ Ò Ò, since Ò ¼ Ò Ò. So if Ñ Ò, we get Ñ Ò and therefore that Ñ Ò. This finally shows that a reduced quadratic irrationality has a purely periodic continued fraction. 2.6 The continued fraction for Æ Example With enough patience one can compute that ¾ ¾ ¾ ¾ ¾ ¾ The example shows a pattern in the continued fraction for the square root. It seems that it repeats itself after encountering ¾ ¼. It also seems to be symmetric around a middle. This is no coincidence: Theorem Let Æ be a natural number, which is not a square. Then where Õ Õ Ò Õ ¾ Õ Ò Æ Õ¼ Õ Õ Ò ¾Õ ¼ Õ Proof. Let Õ ¼ Æ. Then we know that Õ¼ Æ has a purely periodic continued fraction by Theorem Thus Æ Õ ¼ ¾Õ ¼ Õ Õ Ò ¾Õ ¼ Õ. This proves the first statement. Consider now the conjugate Æ Õ¼ of Æ Õ¼. Then Æ Õ¼ Õ Ò Õ Ò Õ ¾Õ ¼ Õ Ò

20 16 CHAPTER 2. CONTINUED FRACTIONS This implies that Æ Õ¼ Õ Ò Õ Ò Õ ¾Õ ¼ Õ Ò By uniqueness of the continued fraction for Æ we get Õ Õ Ò, Õ ¾ Õ Ò. 2.7 A few words on Pells equation This is the diophantine equation Ü ¾ ÆÝ ¾ ( ) where Æ is a natural number, which is not a square. It would be a shame not to mention the elegant way of giving integer solutions to ( ) using the continued fraction expansion of Æ. Using the notation of Theorem 2.6.2, we get (in the usual way) Õ ¼ Æ µ Ò Ò Æ Õ ¼ Æ µøò Ø Ò Multiplying this out leads to Now Ò ÆØ Ò Õ ¼ Ò Ø Ò Ò Õ ¼ Ø Ò Ò Ø Ò Ò Ø Ò ÆØ Ò Õ ¼ Ò µø Ò Ò Ò Õ ¼ Ø Ò µæø ¾ Ò ¾ Ò We conclude that ¾ Ò Æؾ Ò Ò Ø Ò Ò Ø Ò µ µ Ò µ Ò Example Consider the equation Ü ¾ Ý ¾. To find an integer solution we consider the continued fraction expansion for : ¾ ¾ ¾ ¾ and compute the convergents Ø One checks that ¼ ¾ ¾

21 Chapter 3 Exercises 3.1 In class 1. Compute the continued fraction expansion of. 2. Write down the exact steps in an integer algorithm for computing the continued fraction expansion of Æ. 3. Compute the continued fraction expansions of and. 4. Find an integer solution to the equation Ü ¾ Ý ¾. 3.2 Homework 1. Solve all exercises in chapters and ¾ (don t forget the proofs). 2. Invent an integer algorithm to compute the continued fraction expansion for ¾. Compute the first ¼¼ q s in the continued fraction. The sad state of affairs in mathematics is that it is unknown even if the q s are bounded. 1 1 Do this exercise if you have time to spare. 17

22 18 CHAPTER 3. EXERCISES

23 Bibliography [1] C. Pomerance, A tale of two sieves, Notices of the American Mathematical Society 43 (1996),

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

CONTINUED 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. 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 information

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

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

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

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

Algebra 1 Course Title

Algebra 1 Course Title Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept

More information

11 Ideals. 11.1 Revisiting Z

11 Ideals. 11.1 Revisiting Z 11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

More information

Lecture Notes on Polynomials

Lecture Notes on Polynomials Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

More information

Breaking 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 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 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

Integer Factorization using the Quadratic Sieve

Integer Factorization using the Quadratic Sieve Integer Factorization using the Quadratic Sieve Chad Seibert* Division of Science and Mathematics University of Minnesota, Morris Morris, MN 56567 seib0060@morris.umn.edu March 16, 2011 Abstract We give

More information

Primality - Factorization

Primality - Factorization Primality - Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.

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

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

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

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

Zeros of Polynomial Functions

Zeros of Polynomial Functions Review: Synthetic Division Find (x 2-5x - 5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 3-5x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 3-5x 2 + x + 2. Zeros of Polynomial Functions Introduction

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

Lecture 13: Factoring Integers

Lecture 13: Factoring Integers CS 880: Quantum Information Processing 0/4/0 Lecture 3: Factoring Integers Instructor: Dieter van Melkebeek Scribe: Mark Wellons In this lecture, we review order finding and use this to develop a method

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

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

A 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 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 information

How To Prove The Dirichlet Unit Theorem

How To Prove The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

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

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

Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Copy 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 information

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors.

The Prime Numbers. Definition. A prime number is a positive integer with exactly two positive divisors. The Prime Numbers Before starting our study of primes, we record the following important lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) = 1. Lemma (Euclid s Lemma). If gcd(a,

More information

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11. 9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

More information

Math 181 Handout 16. Rich Schwartz. March 9, 2010

Math 181 Handout 16. Rich Schwartz. March 9, 2010 Math 8 Handout 6 Rich Schwartz March 9, 200 The purpose of this handout is to describe continued fractions and their connection to hyperbolic geometry. The Gauss Map Given any x (0, ) we define γ(x) =

More information

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013 FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

More information

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers. Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

More information

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m) Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

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

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.

CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana 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 information

Applications of Fermat s Little Theorem and Congruences

Applications 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 information

A Course on Number Theory. Peter J. Cameron

A Course on Number Theory. Peter J. Cameron A Course on Number Theory Peter J. Cameron ii Preface These are the notes of the course MTH6128, Number Theory, which I taught at Queen Mary, University of London, in the spring semester of 2009. There

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

Factoring Algorithms

Factoring Algorithms Factoring Algorithms The p 1 Method and Quadratic Sieve November 17, 2008 () Factoring Algorithms November 17, 2008 1 / 12 Fermat s factoring method Fermat made the observation that if n has two factors

More information

TEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011

TEXAS A&M UNIVERSITY. Prime Factorization. A History and Discussion. Jason R. Prince. April 4, 2011 TEXAS A&M UNIVERSITY Prime Factorization A History and Discussion Jason R. Prince April 4, 2011 Introduction In this paper we will discuss prime factorization, in particular we will look at some of the

More information

Factoring & Primality

Factoring & Primality Factoring & Primality Lecturer: Dimitris Papadopoulos In this lecture we will discuss the problem of integer factorization and primality testing, two problems that have been the focus of a great amount

More information

Lecture 13 - Basic Number Theory.

Lecture 13 - Basic Number Theory. Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted

More information

Chapter 11 Number Theory

Chapter 11 Number Theory Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

More information

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 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 information

= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that

= 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 information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

PROBLEM SET 6: POLYNOMIALS

PROBLEM SET 6: POLYNOMIALS PROBLEM SET 6: POLYNOMIALS 1. introduction In this problem set we will consider polynomials with coefficients in K, where K is the real numbers R, the complex numbers C, the rational numbers Q or any other

More information

How To Know If A Domain Is Unique In An Octempo (Euclidean) Or Not (Ecl)

How To Know If A Domain Is Unique In An Octempo (Euclidean) Or Not (Ecl) 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

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

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

SECTION 10-2 Mathematical Induction

SECTION 10-2 Mathematical Induction 73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms

More information

On Generalized Fermat Numbers 3 2n +1

On Generalized Fermat Numbers 3 2n +1 Applied Mathematics & Information Sciences 4(3) (010), 307 313 An International Journal c 010 Dixie W Publishing Corporation, U. S. A. On Generalized Fermat Numbers 3 n +1 Amin Witno Department of Basic

More information

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013 Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

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

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

More information

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

More information

Ideal Class Group and Units

Ideal Class Group and Units Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

More information

3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9.

3 0 + 4 + 3 1 + 1 + 3 9 + 6 + 3 0 + 1 + 3 0 + 1 + 3 2 mod 10 = 4 + 3 + 1 + 27 + 6 + 1 + 1 + 6 mod 10 = 49 mod 10 = 9. SOLUTIONS TO HOMEWORK 2 - MATH 170, SUMMER SESSION I (2012) (1) (Exercise 11, Page 107) Which of the following is the correct UPC for Progresso minestrone soup? Show why the other numbers are not valid

More information

Vocabulary Words and Definitions for Algebra

Vocabulary Words and Definitions for Algebra Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

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

4.2 Euclid s Classification of Pythagorean Triples

4.2 Euclid s Classification of Pythagorean Triples 178 4. Number Theory: Fermat s Last Theorem Exercise 4.7: A primitive Pythagorean triple is one in which any two of the three numbers are relatively prime. Show that every multiple of a Pythagorean triple

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

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

Math Workshop October 2010 Fractions and Repeating Decimals

Math 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 information

An Overview of Integer Factoring Algorithms. The Problem

An Overview of Integer Factoring Algorithms. The Problem An Overview of Integer Factoring Algorithms Manindra Agrawal IITK / NUS The Problem Given an integer n, find all its prime divisors as efficiently as possible. 1 A Difficult Problem No efficient algorithm

More information

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION

ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION ALGEBRAIC APPROACH TO COMPOSITE INTEGER FACTORIZATION Aldrin W. Wanambisi 1* School of Pure and Applied Science, Mount Kenya University, P.O box 553-50100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Computing exponents modulo a number: Repeated squaring

Computing exponents modulo a number: Repeated squaring Computing exponents modulo a number: Repeated squaring How do you compute (1415) 13 mod 2537 = 2182 using just a calculator? Or how do you check that 2 340 mod 341 = 1? You can do this using the method

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

GENERATING SETS KEITH CONRAD

GENERATING SETS KEITH CONRAD GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

More information

Unique Factorization

Unique Factorization Unique Factorization Waffle Mathcamp 2010 Throughout these notes, all rings will be assumed to be commutative. 1 Factorization in domains: definitions and examples In this class, we will study the phenomenon

More information

SOLVING POLYNOMIAL EQUATIONS

SOLVING POLYNOMIAL EQUATIONS C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

More information

An Innocent Investigation

An Innocent Investigation An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

More information

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

More information

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS (Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

More information

An Introductory Course in Elementary Number Theory. Wissam Raji

An Introductory Course in Elementary Number Theory. Wissam Raji An Introductory Course in Elementary Number Theory Wissam Raji 2 Preface These notes serve as course notes for an undergraduate course in number theory. Most if not all universities worldwide offer introductory

More information

Elementary factoring algorithms

Elementary factoring algorithms Math 5330 Spring 013 Elementary factoring algorithms The RSA cryptosystem is founded on the idea that, in general, factoring is hard. Where as with Fermat s Little Theorem and some related ideas, one can

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

Factoring Polynomials

Factoring Polynomials Factoring Polynomials Hoste, Miller, Murieka September 12, 2011 1 Factoring In the previous section, we discussed how to determine the product of two or more terms. Consider, for instance, the equations

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

Factorization Methods: Very Quick Overview

Factorization Methods: Very Quick Overview Factorization Methods: Very Quick Overview Yuval Filmus October 17, 2012 1 Introduction In this lecture we introduce modern factorization methods. We will assume several facts from analytic number theory.

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

Algebra Practice Problems for Precalculus and Calculus

Algebra Practice Problems for Precalculus and Calculus Algebra Practice Problems for Precalculus and Calculus Solve the following equations for the unknown x: 1. 5 = 7x 16 2. 2x 3 = 5 x 3. 4. 1 2 (x 3) + x = 17 + 3(4 x) 5 x = 2 x 3 Multiply the indicated polynomials

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

MACM 101 Discrete Mathematics I

MACM 101 Discrete Mathematics I MACM 101 Discrete Mathematics I Exercises on Combinatorics, Probability, Languages and Integers. Due: Tuesday, November 2th (at the beginning of the class) Reminder: the work you submit must be your own.

More information

8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

More information

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005

Polynomials. Dr. philippe B. laval Kennesaw State University. April 3, 2005 Polynomials Dr. philippe B. laval Kennesaw State University April 3, 2005 Abstract Handout on polynomials. The following topics are covered: Polynomial Functions End behavior Extrema Polynomial Division

More information

The Ideal Class Group

The Ideal Class Group Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned

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

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

More information

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

More information

Basic Algorithms In Computer Algebra

Basic Algorithms In Computer Algebra Basic Algorithms In Computer Algebra Kaiserslautern SS 2011 Prof. Dr. Wolfram Decker 2. Mai 2011 References Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, 1993. Cox, D.; Little,

More information

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may

Number Theory. Proof. Suppose otherwise. Then there would be a finite number n of primes, which we may Number Theory Divisibility and Primes Definition. If a and b are integers and there is some integer c such that a = b c, then we say that b divides a or is a factor or divisor of a and write b a. Definition

More information

Primality Testing and Factorization Methods

Primality Testing and Factorization Methods Primality Testing and Factorization Methods Eli Howey May 27, 2014 Abstract Since the days of Euclid and Eratosthenes, mathematicians have taken a keen interest in finding the nontrivial factors of integers,

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov

Primes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES

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

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

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2

QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 QUADRATIC RECIPROCITY IN CHARACTERISTIC 2 KEITH CONRAD 1. Introduction Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law in F[T ] (see [4, Section 3.2.2] or [5]) lets

More information