CONTINUED FRACTIONS AND FACTORING. Niels Lauritzen


 Harold Higgins
 1 years ago
 Views:
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 FermatKraitchik 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 nontrivial 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 FermatKraitchik 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 nontrivial 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 ÆØ¾ 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 Òth convergent of a continued fraction Ü Ü ¾ Ü by Ü Ü ¾ Ü Ò. Then the Òth 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 Òth 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 Òth 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) É¾ 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 ÕÉµ µ È ÕÉµ 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 Òth 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 ÆØ¾ 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
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 informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationCHAPTER 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 informationContinued Fractions. Darren C. Collins
Continued Fractions Darren C Collins Abstract In this paper, we discuss continued fractions First, we discuss the definition and notation Second, we discuss the development of the subject throughout history
More informationAlgebra 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 information11 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 informationLecture 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 informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationMATH10040 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 informationMath Review. for the Quantitative Reasoning Measure of the GRE revised General Test
Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important
More informationInteger 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 informationElementary 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 informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationit 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 informationZeros 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 informationI. 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 informationZeros of Polynomial Functions
Review: Synthetic Division Find (x 25x  5x 3 + x 4 ) (5 + x). Factor Theorem Solve 2x 35x 2 + x + 2 =0 given that 2 is a zero of f(x) = 2x 35x 2 + x + 2. Zeros of Polynomial Functions Introduction
More informationPrimality  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 informationLecture 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 informationStanford Math Circle: Sunday, May 9, 2010 SquareTriangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 SquareTriangular 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 informationMath 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 informationCopy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.
Algebra 2  Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers  {1,2,3,4,...}
More information9. 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 n1 x n1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role
More informationMath 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 informationMarch 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 informationHomework 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 informationThe 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 informationCHAPTER 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 informationInternational Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013
FACTORING CRYPTOSYSTEM MODULI WHEN THE COFACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II MohammediaCasablanca,
More informationCHAPTER 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 informationInteger 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 informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREERREADY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationChapter 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 informationAPPLICATIONS 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 informationa 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
More information2.3. 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 informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationFactoring & 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 informationLecture 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 nonnegative integers. We say that A divides B, denoted
More information3. 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 informationMath Workshop October 2010 Fractions and Repeating Decimals
Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On wellordering and induction: (a) Prove the induction principle from the wellordering principle. (b) Prove the wellordering
More informationSubsets 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 informationWelcome 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 informationVocabulary 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 informationSECTION 102 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 informationOn 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 informationPROBLEM 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 informationIdeal 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 informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationJUST 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 informationTEXAS 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 informationThis 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 informationApplications of Fermat s Little Theorem and Congruences
Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4
More informationThnkwell 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 informationFactoring 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 informationLectures 56: 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 informationSOLVING 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 information4.2 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 informationUnique 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 informationElementary 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 informationGENERATING 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 information3. Applications of Number Theory
3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a
More information8 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 informationOn 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 informationFactoring 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 informationComputing 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 informationALGEBRAIC 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 55350100, Kakamega, Kenya. Shem Aywa 2 Department of Mathematics,
More informationMACM 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 information6 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 informationSECTION 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 informationFACTORING 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 informationAlgebra 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 informationparent 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 informationa 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 informationAn 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 informationGREATEST COMMON DIVISOR
DEFINITION: GREATEST COMMON DIVISOR The greatest common divisor (gcd) of a and b, denoted by (a, b), is the largest common divisor of integers a and b. THEOREM: If a and b are nonzero integers, then their
More informationThe last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving NonConditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
More informationLecture 3: Finding integer solutions to systems of linear equations
Lecture 3: Finding integer solutions to systems of linear equations Algorithmic Number Theory (Fall 2014) Rutgers University Swastik Kopparty Scribe: Abhishek Bhrushundi 1 Overview The goal of this lecture
More informationMATH 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 informationSECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS
SECTION 2.5: FINDING ZEROS OF POLYNOMIAL FUNCTIONS Assume f ( x) is a nonconstant polynomial with real coefficients written in standard form. PART A: TECHNIQUES WE HAVE ALREADY SEEN Refer to: Notes 1.31
More informationDiscrete 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 informationMath Circle Beginners Group October 18, 2015
Math Circle Beginners Group October 18, 2015 Warmup problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd
More informationPolynomials. 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 informationPrimes 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 informationThe 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 informationModule 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 informationCommon Core Unit Summary Grades 6 to 8
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity 8G18G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More information3 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 informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS 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 the identity, such that (i) the
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationMTH124: Honors Algebra I
MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,
More informationFlorida Math for College Readiness
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourthyear math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
More informationSouth Carolina College and CareerReady (SCCCR) Algebra 1
South Carolina College and CareerReady (SCCCR) Algebra 1 South Carolina College and CareerReady Mathematical Process Standards The South Carolina College and CareerReady (SCCCR) Mathematical Process
More information