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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.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ----------------------------------------------------------------------------
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ABSTRACT ----------------------------------------------------------------------------------------------
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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 -------------------------------------------------------------------------
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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 cosx  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  pq  qp
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  ab  ba
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 gcds1 , 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 gcds1 , 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 .
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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 

in

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   nk1 


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   nk  k n . By Theorem 2.4 since k  P P ,

k   k so nk   nk  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   nk1 


thus by

r ai
i 1 q
i

 .

2) Let k  P P , p  P . Consider  pk   pk  k p . Since by Theorem 2.4 k   k we can substitute
and get  pk   pk  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))
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answer += i + "' = " + getNumberDerivative(i) + "\r\n";
myTextArea.setText(answer);
}
}
}
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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.
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