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FOUNDATIONS FOR NUMBER ANALYSIS By Martin Storm A SENIOR RESEARCH PAPER PRESENTED TO THE DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE OF STETSON UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE STETSON UNIVERSITY 2005 ACKNOWLEDGMENTS I would like to acknowledge Robert Lamar, for being a good roommate and sounding board for the ideas I had. I would also like to thank Andrew Moedinger for writing a very useful program for gathering data. Furthermore, I would like to thank my other roommates, Nathan Edwards and Chris Shaw for being good roommates while I was working on this project. And finally, I would like to thank Hari Pulapaka for being a great Advisor and not getting too frustrated with me during the course of this project. 2 TABLE OF CONTENTS ACKNOWLEDGEMENTS ---------------------------------------------------------------------------- 2 ABSTRACT ---------------------------------------------------------------------------------------------- 4 CHAPTERS 1. Preliminaries ----------------------------------------------------------------------------------------1.1. Introduction -----------------------------------------------------------------------------------1.1.1. A Brief History of Calculus -------------------------------------------------------1.1.2. A Shorter History of the Number Derivative -----------------------------------1.2. Notation and Terminology ------------------------------------------------------------------1.3. Known Results --------------------------------------------------------------------------------1.3.1. A Formula for the Number Derivative -------------------------------------------1.3.2. The Quotient Rule ------------------------------------------------------------------2. Calculus Inspired Results --------------------------------------------------------------------------2.1. Derivative Rules ------------------------------------------------------------------------------2.1.1. Power Rule ---------------------------------------------------------------------------2.2. Parallels to e x ---------------------------------------------------------------------------------3. Problems and Conjectures ------------------------------------------------------------------------REFERENCES ----------------------------------------------------------------------------------------APPENDIX A: NumberDerivative by Andrew Moedinger --------------------------------------BIOGRAPHICAL SKETCH ------------------------------------------------------------------------- 3 5 5 5 6 6 7 7 7 8 8 8 8 13 15 16 20 ABSTRACT FOUNDATIONS FOR NUMBER ANALYSIS By Martin Storm May 2006 Advisor: Hari Pulapaka Department: Mathematics and Computer Science Classical Calculus may be categorized into Differential Calculus and Integral Calculus. In Differential Calculus, the derivative of a function is introduced as a method of measuring the rate of change and growth of functions. Recently, a new number-theoretic analogue for a derivative was defined in a paper by Westrick [2]. The number derivative of a positive integer depends on its prime-decomposition. This paper is a continued study of the number derivative with an emphasis on results motivated by theorems from Classical Calculus. 4 CHAPTER 1 Preliminaries 1.1. Introduction 1.1.1. A Brief History of Calculus The roots of Calculus extend back to the time of the Greeks, and were used to obtain approximations for various Real Numbers in a “method of exhaustion” developed supposedly by Eudoxus of Cnidus [5]. This method was then used by Archimedes to obtain an approximation for the number [4]. The next two big names in the history of Calculus are that of Kepler and Cavalieri, who lived in the 16th century. Kepler was responsible for considering and attempting to find a formula for the area of an ellipse by attempting to sum up the lengths of lines [4] while Cavalieri considered volumes generated by rotating shapes [5]. After Kepler and Cavalieri, the early years of calculus begin with Newton and Leibniz [5]. In 1669 and 1671, Newton wrote two papers which due to unfortunate circumstances would not be published until some 40 or 50 years later. In these two papers, Newton introduced the series form for both sin x and cosx as well as introducing an early form of ex [4]. Leibniz on the other hand was able to publish his two papers in 1684 and 1686, and they introduced much of common calculus notation today, specifically the use of dx/dy as a symbol for differentiation of a function in x with respect to with respect to y and integration as well as , the symbol for integration [4]. The next significant period of calculus was still influenced by both Newton and Leibniz, but also saw contributions from Fermat, Bernoulli and Rolle [5]. In this period, Leibniz introduces a method for finding minima and maxima of a function [4], something done by Fermat as well [4]. Also, Rolle states Rolle’s Theorem which is simply the Mean Value Theorem when the derivative is zero. Also during this time period Bernoulli discovers and sells L’hôpital’s rule and Leibniz discovers the product and quotient rules [5]. And finally, some of the last few players were Riemann and Heine in the late 5 1800’s [5]. In this time period Heine introduces the Epsilon-Delta proof and Riemann introduces his Integral. 1.1.2. A Shorter History of the Number Derivative. In 2002 at the 14th Summer Conference of the International Tournament of Towns a number-theoretic concept called the number derivative was proposed. After this concept was proposed it was little over a year before a paper titled “Investigations of the Number Derivative” was written by Linda Westrick under the direction of Pavlo Pylyavaskyy from MIT [2]. This paper covered many topics concerning the number derivative, including but not limited to extending it from the positive integers to the rational numbers and finding bounds on the number derivative. 1.2. Terminology and Notation. For ease of use and reference in this work, we denote the set of prime numbers as P and the set P P is used to denote the set of numbers whose members are a prime number raised to itself. Furthermore, we use the symbol S to denote the cardinality of (or number of elements in) a set S. Additionally we use Z to represent the set of positive integers. Lastly, remember that any real number that is a solution to any polynomial equation with integer coefficients is a member of the set of Algebraic Numbers. 1.2.1 Definition of the Number Derivative We define the number derivative as a function ‘ : Z Z as follows: 1) If p P then p 1 2) If n pq then n pq qp 3) 0 0 Example: 6 2 * 3 2 * 3 3 * 2 3 2 5 . 6 This example illustrates the fact that for any two primes p and q, pq p q . Lemma 1.1: 1 0 Proof: Let n Z . Then the following holds: n 1 n 1 n n 1 n 1 n 0 n 1 1 0 1.3. Known Results In this section we show some previous results concerning the number derivative as they have an impact on this paper either because they are used or because they have an analogy in classical calculus. 1.3.1 Formula for the Number Derivative This formula comes from [2]. It provides a simple formula for calculating the number derivative if the prime factorization of the number in question is known. Theorem 1.1: If n r i 1 piai with r , ai N and pi P then n n r i 1 ai . pi Proof: See [2] 1.3.2 The Quotient Rule. This theorem also is a result from [2] and is analogous to the Quotient Rule from Classical Calculus. a ab ba Theorem 1.2: Let a, b N . Then . b2 b Proof: See [2] 7 CHAPTER 2 Calculus Inspired Results As the primary focus of this work is to examine at the number derivative and try and find analogies between it and the classical derivative, it serves to first explore which of the rules for differentiation still hold. 2.1. Derivative Rules 2.1.1 The Power Rule One of the first differentiation rules that a calculus student learns is the power rule, and given the definition of the number derivative, the power rule is actually the most likely to be able to function correctly. And in this case, it does. Theorem 2.1: Let n m k where n, m, k N . Then n' kmk 1m Proof: Let n m k and let m n k p ai i 1 i r r i 1 r piai with r , ai N and pi P . Then by substitution i 1 pikai . Now by Theorem 1 from [2] we know n n n n kai i 1 p i r kn k r i 1 r i 1 k r k r ai pi p ikai p ai i 1 i p ai i 1 i r i 1 k ai pi r i 1 k 1 kmk 1 m 8 ai pi p ai i 1 i r ai i 1 p i r r i 1 kai Thus: pi 2.2 Parallels to ex One of the more interesting functions in calculus happens to be f x e x since it happens to be its own derivative. It would be nice to prove that there is a number say k that is it’s own derivative, like ex. Fortunately, there are such numbers, but before we can prove this, we first need a lemma. Lemma 2.1: Let S be a set whose elements are pairwise relatively prime. Then if ai Z 0 1 i S S ai i i s 1 with si S then for any solution in ai the following holds true: 1) ! ak 0 and ai, i k ai 0 2) ak sk Proof: The proof is by induction on S Base Case: S 2 In this case gcds1 , s2 1 and a1 a2 1 with a1 , a2 Z 0. Now we can say the following: s1 s2 a1 s 2 a 2 s1 s1 s 2 a1 s 2 s1 s 2 a 2 s1 a1 s 2 s1 s 2 a 2 Thus a1s2 | s1 s2 a2 and therefore if a1 0 , then s2 | s1 ( s2 a2 ) . Now remember that since gcds1 , s2 1 , by Euclid’s Lemma [3] we then know that s2 | s2 a2 . This means that s2 a2 s2 . But since we know that a2 0 this implies that s2 s2 a2 which forces s2 s2 a2 a2 0 . 9 If, however, a1 0 then a1 1 and thus a1 s1 . s1 Induction Hypothesis: For S i 1 ai 1 with 1 i S , ai Z 0 and 1 i, j S , gcd si , s j 1 implies that si only one ai non-zero and that ai si . Induction Step: Try k + 1. Thus k 1ai i 1 si . Note that if a, b and c are all pairwise relatively prime then ab is relatively prime to c. By this we can then say that sksk+1 is relatively prime to the rest of the si’s. Now observe that ak ak 1 ak sk 1 ak 1sk . Thus we can now say: sk sk 1 sk sk 1 ak sk 1 ak 1sk 1. So if a' j ai and s ' j si 1 i k 1 and 1 j k 1 with i 1 s sk sk 1 i k 1ai a'k ak sk 1 sk ak 1 and s 'k sk sk 1 we then say k j 1 a' j s' j 1 and then the Induction Hypothesis finishes the proof. Now we are ready to find an analogy for f x e x . Theorem 2.2: Let k Z then k ' k if and only if k P P . Proof: From Theorem 1 of the Westrick Paper [2] we know k' k p ai i cancellation we can say p ai i k . Thus by i 1 . Now we apply Lemma 1 to show that ! ai 0 which we will i denote an. This means that k in pi0 pna n pna n . But Lemma 1 also tells us that an pn so thus k pnan P P . 10 Let k P P . Thus p P k p p . By Theorem 1 of the Westrick Paper [2] we can say: k ' p p ' p p p p p k . p As a result of this theorem, we then call numbers which are their own derivative p p s. While it’s nice to have an analogy to f x e x for the number derivative, it would also be useful if it held some of the same properties. Theorem 2.3: Let n Z , p P , k P P with n r q ai i 1 i where 1 i r , qi P , ai Z . The following properties hold: 1) nk nk1 r ai i 1 q i 2) pk k p 1 3) ki n i 1 n n k i 1 i Proof: 1) Let n Z , k P P with n r q ai i 1 i where 1 i r with qi P , ai Z . By the definition of the number derivative nk nk k n . By Theorem 2.4 since k P P , k k so nk nk nk k n n . Then by Theorem 1 from [2] n n substitution we get nk k n n r ai i 1 q i r ai i 1 q i . Factoring results in nk nk1 thus by r ai i 1 q i . 2) Let k P P , p P . Consider pk pk k p . Since by Theorem 2.4 k k we can substitute and get pk pk pk . Since p P we then say pk k pk k ( p 1) . 11 3) Proof by Induction on n. Base Case n = 1: ki 1 i 1 k k 1 i k i 1 i Induction Hypothesis: ki n i 1 n n k i 1 i Induction Step: Consider the (n + 1)th case. n 1 i ki k n 1 k n 1 k i i 1 n ki k n 1 i 1 n k k i i 1 n 1 k i 1 i k n 1 n k i 1 i k n 1 n k i 1 i k n n 1 k n 1 n k i 1 i n 1 n 1 n This completes the proof. 12 n n 1 k i 1 i n k i 1 i CHAPTER 3 Problems and Conjectures Although this paper starts to draw some parallels between the standard derivative and the number derivative, there are still many problems left to be considered regarding the number derivative. For example, by just considering the formula for the number derivative presented in [2], it should be possible to take the derivative of numbers such as 3 3 . Incidentally, this ends up being 3 3 3 2 3 1 3 3 3 3 2 3 2 3 2 3 . 3 2 2 Thus it might be possible to remove the restriction that we should be using integer powers of primes. In that case, we could allow the use of rational powers, and still be able to compute a number derivative. Once this is done, if we additionally had a rule for taking the derivative of two numbers added together, we would then be able to take the derivative of an algebraic number. Furthermore, while Theorem 2.2 states that the only a prime raised to itself is its own derivative we still have some interesting possibilities. If we remove the integer power restriction this Theorem ceases to hold. The question then becomes: Let n s a i 1 i s i 1 piai . If we restrict k , does there then exists a largest n regardless of what s is such that n is its own derivative? And if so, how does the derivative behave for numbers greater then that n? Of course, another problem would be to consider the equation a b a b . Interestingly enough there are solutions to this, such as 4 35 39 16 4 12 5 35 and 29 26 55 16 1 15 29 26 . The question here would be, which relatively prime pairs of a and b does this happen with, and can they be classified? 13 Additionally, since the standard derivative can be used to measure things about the original function it is only natural to ask if there is an interpretation for the number derivative. More simply put, the question would be “What, if anything, does the number derivative tell us about the original number?” Obviously, if we have a large derivative it is symptomatic of either a few large primes or the original number is fairly composite. Another associated question in this instance would be “How do you tell whether or not a number has a large number of factors or a few large prime factors?” 14 REFERENCES [1] James Stewart, Calculus: Concepts and Contexts, Brooks/Cole, 2001. [2] Linda Westrick, Investigations of the Number Derivative, 2003. [3] Kenneth H. Rosen, Elementary Number Theory and it’s applications, Fifth Ed., Pearson Addison Wesley, 2005. [4] J J O’Conner and E F Robertson “A History of the Calculus” Calculus History, 4 April 2005. http://www. -groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html. [5] Donald Lacon Jr. “Timeline of Calculus” A brief history of calculus, 4 April 2005. <http://www.meta-religion.com/Mathematics/Articles/timeline_of_calculus.htm>. 15 APPENDIX A. NumberDerivative by Andrew Moedinger ////////////////////////////////////////////////// // // //Class: NumberDerivative.java // //Author: Andrew Moedinger // //Purpose: To calculate number derivatives.// // // //Date: Jan 26, 2005 // // // ////////////////////////////////////////////////// import java.awt.*; import java.awt.event.*; import java.applet.Applet; import java.math.BigInteger; public class NumberDerivative extends Applet implements ActionListener { private int [] myPrimes; private Button mySubmitButton; private TextField myMinField; private TextField myMaxField; private TextArea myTextArea; private Label myLabel1, myLabel2; private Label myTitle; private Label myAuthorLabel1, myAuthorLabel2; public void init () { mySubmitButton = new Button("Find Number Derivatives"); myMaxField = new TextField("",5); myMinField = new TextField("",5); myLabel1 = new Label("Start Number"); myLabel2 = new Label("End Number"); myTextArea = new TextArea(27,20); myTitle = new Label("Number Derivatives"); myAuthorLabel1 = new Label("Created © 2005"); myAuthorLabel2 = new Label("Andrew Moedinger"); myTitle.setForeground(new Color(0,0,255)); myAuthorLabel1.setForeground(new Color(0,0,255)); myAuthorLabel2.setForeground(new Color(0,0,255)); myMaxField.setBackground(Color.WHITE); myMinField.setBackground(Color.WHITE); myLabel1.setBackground(Color.WHITE); myLabel2.setBackground(Color.WHITE); myTextArea.setBackground(Color.WHITE); myTitle.setBackground(Color.WHITE); myAuthorLabel1.setBackground(Color.WHITE); 16 myAuthorLabel2.setBackground(Color.WHITE); this.add(myTitle); this.add(myLabel1); this.add(myMinField); this.add(myLabel2); this.add(myMaxField); this.add(mySubmitButton); this.add(myTextArea); this.add(myAuthorLabel1); this.add(myAuthorLabel2); this.setSize(200,600); this.setBackground(Color.WHITE); mySubmitButton.addActionListener(this); } public BigInteger getNumberDerivative(BigInteger num) { BigInteger tnum = new BigInteger(num.toString()); BigInteger factor; BigInteger numerator, denominator; //long denominator, numerator; numerator = BigInteger.ZERO; denominator = BigInteger.ONE; // by definition 1', 0' and p' where is a prime number = 1 if (num.equals(BigInteger.ONE) || num.equals(BigInteger.ZERO) || isPrime(num)) return BigInteger.ONE; // find each factor f and add factor as 1/f while (tnum.compareTo(BigInteger.ONE) > 0) { factor = getNextFactor(tnum); if (factor.compareTo(BigInteger.ONE) <= 0) break; else { tnum = tnum.divide(factor); // we are adding 1/factor + the current result of numerator/denominator // we do this by (numerator*factor)/(denominator*factor) + 1*(denominator)/(factor*denominator) // thus, numerator = (numerator*factor) + (denominator)*1 // and, denominator = denominator*factor // numerator = numerator.multiply(new BigInteger(String.valueOf(factor))).add(denominator); denominator = denominator.multiply(new BigInteger(String.valueOf(factor))); } } 17 // multiply sum of each 1/f by the number numerator = numerator.multiply(num); if (!numerator.mod(denominator).equals(BigInteger.ZERO)) System.out.println("numerator % denominator != 0, thus result not an integer!!!"); return numerator.divide(denominator); } // returns the next smallest factor other than 1 // or the number itself if there is no other factor public BigInteger getNextFactor(BigInteger num) { BigInteger i; BigInteger two = new BigInteger("2"); for (i = new BigInteger("2"); i.compareTo(num.divide(two)) <= 0; i = i.add(BigInteger.ONE)) { if (num.mod(i).equals(BigInteger.ZERO)) return i; } return num; } // simple primality check public boolean isPrime(BigInteger num) { BigInteger i; BigInteger two = new BigInteger("2"); if (num.compareTo(BigInteger.ONE) <= 0 || num.mod(two).equals(BigInteger.ZERO)) return false; for (i = new BigInteger("3"); i.compareTo(num.divide(two)) <= 0; i = i.add(two)) { if (num.mod(i).equals(BigInteger.ZERO)) return false; } return true; } // upon any action, the number derivatives will be calculated for the min and max values // specified in the input fields public void actionPerformed(ActionEvent e) { if (e.getSource() == mySubmitButton) { BigInteger i, min, max; min = new BigInteger(myMinField.getText()); max = new BigInteger(myMaxField.getText()); String answer = ""; for (i = min; i.compareTo(max) <= 0; i = i.add(BigInteger.ONE)) 18 answer += i + "' = " + getNumberDerivative(i) + "\r\n"; myTextArea.setText(answer); } } } 19 BIOGRAPHICAL SKETCH Martin Storm was born on the 18th of November 1983 in St. Petersburg Florida. In 1987 his family moved to Lexington Massachusetts, and after nine years moved to Yorba Linda California. In 2002 he graduated from Esperanza High School, and made the decision to attend Stetson University. While at Stetson University he has participated in the Putnam Exam and also an ACM competition. He is currently a Mathematics Major, intent on graduating at the end of the Spring Semester of 2006. 20