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Contents

1 GENERAL INFORMATION 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Text Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Course Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 Internet access and unit homepage . . . . . . . . . . . . . . . . . . . 2

1.5 Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.6 Tutorial Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.7 Calculators and Computers . . . . . . . . . . . . . . . . . . . . . . . 3

1.8 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.9 Plagiarism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.10 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.11 Administrative and General Enquiries . . . . . . . . . . . . . . . . . 4

1.12 Academic Enquiries . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 ASSIGNMENT SCHEDULE 5

3 COURSE OUTLINE (using the fifth edition) 6

4 COURSE OUTLINE (using the fourth edition) 8

5 COURSE OUTLINE (using the third edition) 10

6 ASSIGNMENTS 12

7 NUMBERS: A NUMBER THEORETIC CALCULATOR 19

i

PMTH 338NUMBER THEORY

1 GENERAL INFORMATION

1.1 Background

This unit on number theory is elementary in the sense that it uses little from other

mathematics units that you have done. The main prerequisites are contained in the

first year mathematics course and in the second year linear algebra course. It should

be noted however that number theory is a very large subject with a history extending

back to the beginning of recorded civilisation. It would be possible to design a third

year level course on algebraic number theory which would have abstract algebra

as a prerequisite, and another course on analytic number theory which would have

complex analysis as a prerequisite. It is hoped that the present course will convey the

flavour of number theory while still being available to as many third year students

as possible.

In recent years, number theory has found important applications, and some of

these are discussed in the course. The applications of number theory to computer

science are particularly important. However, no background in computer science is

necessary to take this course. The text book contains a good deal of discussion on

the connections with computing for those who may wish to read a little more in this

direction.

Lectures to internal students will be at the rate of approximately five per fort-

night, with one tutorial per fortnight.

1.2 Text Book

The recommended text book is Elementary Number Theory and Its Applications

by Kenneth H. Rosen (Addison-Wesley), 3rd edition 1993, 4th edition 2000, or 5th

edition 2005.

1.3 Course Structure

The course is divided into six sections of approximately two weeks study per section.

1

The material in the first half of the course is relatively straightforward, but there

is some more difficult material in the second half. It is important to have a good

grasp of the first half of the course before you try the second half, and for most

students this requires time to allow the ideas and methods to sink in. In other

words, it is not a good idea to leave it to the last week to start working properly on

the course.

1.4 Internet access and unit homepage

There is a unit homepage with a Bulletin Board (for discussions and questions)

under the address

http://turing.une.edu.au/˜pmth338

Actual informations concerning the unit will be announced through this home-

page and sent out by email. Therefore it is strongly recommended that you check

the unit homepage regularly and that you provide an email address which you also

check regularly.

1.5 Assignments

There is one assignment for each section, and you are expected to hand in solutions

to each assignment. Solutions will be returned with your assignment (for external

students).

The assignments contribute 30% towards the assessment. Nevertheless, the as-

signments are intended primarily as a learning mechanism. Some of the examination

questions will contain sections which are similar to assignment questions.

Most of the assignment questions should come out by following procedures listed

in the book. However some questions are more difficult. If you have tried hard but

cannot make any further progress with a problem, go on to the next one. When the

solutions are returned, try the problem again, using the first part of the solution as

a hint. Number theory is notorious as a subject where a problem may look easy but

the solution can be difficult.

2

1.6 Tutorial Problems

To do a subject such as number theory, you need to solve a fairly large number of

problems, more than just those contained in the assignments. The list of tutorial

problems is intended to fill this need. I suggest you do a number of the tutorial

problems as you study your lecture notes and try the remainder later on. Some of the

examination questions will contain sections which are similar to tutorial questions.

Solutions to the tutorial problems are included in a separate booklet entitled Tutorial

Problems.

1.7 Calculators and Computers

A few assignment questions require use of a calculator, and in the solutions I will

try to indicate when use of a calculator is appropriate. In general though you should

always try to do an assignment question using just pencil and paper, as this will force

you to look for the most efficient method of doing a numerical problem. Sometimes

there may be a long and inefficient way of doing a problem as well as a short and

efficient way.

Also included in this booklet is a description of the number theoretic program

NUMBERS and some exercises suitable for use with this program. You need a

computer to use NUMBERS, and this package can be thought of as a super calculator

for use in number theory.

1.8 Assessment

Assessment will be by one three-hour examination counting 70% (held in the June

examination period), and by assignments counting 30% as indicated previously. The

whole semester’s work as described in the course outline is examinable.

1.9 Plagiarism

Students are warned to read the statements regarding the University’s Policy on

Plagiarism as set out in the following documents:

The University of New England Academic Board Policy on Plagiarism and Aca-

demic Misconduct: Coursework

3

http://www.une.edu.au/policies/pdf/plagiarismcoursework.pdf,

the document Avoiding Plagiarism and Academic Misconduct (Coursework)

http://www.une.edu.au/policies/pdf/plagiarismstudentinfocw.pdf,

the relevant sections on plagiarism provided in the UNE Referencing Guide

http://www.une.edu.au/tlc/referencing.pdf.

1.10 General

Number theory is one of the most enjoyable subjects in mathematics and many peo-

ple find that they develop a deeper understanding of mathematics through studying

number theory. Many of the famous mathematicians of the past have their names

attached to theorems in number theory. Perhaps the most famous problem in all

of mathematics is Fermat’s Last Theorem. In the final section of this course you

will see some special cases of the theorem. The unit closes with a brief history of

Fermat’s Last Theorem leading up to the recent solution of this problem.

1.11 Administrative and General Enquiries

For all administrative matters and information relating to your enrolment and can-

didature, please contact the UNE Student Center:

Email: [email protected]

Call: 02 6773 4444

Fax: 02 6773 4400

Mail: Student Centre

The University of New England

ARMIDALE, NSW 2351

1.12 Academic Enquiries

For academic enquiries concerning the content and assessment requirements of this

unit, contact the unit coordinator.

4

Dr Gerd Schmalz

Lecturer and Unit Coordinator

PMTH 338 Number Theory

School of Mathematical and Computer Sciences

University of New England

Armidale NSW 2351

Phone: 02 6773 3182

Fax: 02 6773 3312

Email: [email protected]

2 ASSIGNMENT SCHEDULE

Assignment Last date for submitting/posting

1 5 March

2 19 March

3 30 April

4 14 May

5 21 May

6 4 June

REQUESTS FOR AN EXTENSION OF TIME FOR SUBMITTING

AN ASSIGNMENT ARE TO BE ADDRESSED TO THE UNIT

COORDINATOR BEFORE THE DUE DATE. According to the FAS

policy, penalties apply for late assignments without approval.

(These dates for internals are subject to change. Any changes will be announced

in lectures.)

5

3 COURSE OUTLINE (using the fifth edition)

For those students who have the new fifth edition of Rosen.

TEXT BOOK REFERENCES TO ROSEN (5th edition.)

Lecture 1 Algebraic properties of Numbers p. 577-579

Lecture 2 The Well-Ordering Property p. 6, p.23-27

Lecture 3 Divisibility p. 37-39

Lectures 4, 5 Prime numbers pp. 67–72

Prime Number Theorem pp. 77–83

Long runs of composites p. 82

Greatest common divisors pp. 90–94

The Euclidean algorithm pp. 97–105

Lecture 6 The fundamental theorem of arithmetic pp. 108–110

Lecture 7 Infinitely many primes of the form 4n + 3 p. 114

Fermat factorisation pp. 123–128

Lecture 8 Linear Diophantine equations pp. 133–137

END OF MATERIAL FOR ASSIGNMENT 1

Lecture 9 Congruences pp. 141–149

Lecture 10,11 Linear Congruences pp. 153–156

Lecture 12 The Chinese Remainder Theorem pp. 158–161

Lecture 13 The Calendar pp. 195–199

Round-Robin tournaments pp. 200–202


Lecture 14 Wilson’s theorem pp. 215–217

Fermat’s little theorem pp. 217–219

Lecture 15 Euler’s theorem pp. 233–236

Lecture 16 Multiplicative functions pp. 239–256

Lecture 17 Sum and number of divisors pp. 250–253

Modular Exponentiation pp. 147–149

Lectures 18, 19 Public-key Cryptography pp. 277–284,

pp. 305–315


6

Lectures 20 Primitive roots pp. 333–339

Lectures 21 Primitive roots for primes pp. 341–343

Lecture 22 Index Arithmetic pp. 355–363


Lecture 23 Quadratic Residues pp. 401–403

Lecture 24 Quadratic Residues and the Legendre Symbol pp. 404–410

Lecture 25 Quadratic reciprocity pp. 417–426

(You only need to know the statement of the result and

how to use it. Proofs will not be examined).

Lecture 26 The Jacobi symbol pp. 430–436

No proofs in this lecture will be examined.

Concentrate on the numerical problems.

Lecture 27 Pseudoprimes pp. 223–231

Euler Pseudoprimes pp. 439–446


Lectures 28, 29 Pythagorean triples pp. 510–514

Lectures 30, 31 Some Diophantine equations pp. 520–522

Lecture 32 History of Fermat’s Last Theorem pp. 516–520


7

4 COURSE OUTLINE (using the fourth edition)

For those students who have the fourth edition of Rosen.

TEXT BOOK REFERENCES TO ROSEN (4th edition.)

We will assume that you are already familiar with the background material from

Appendix A and Chapter 1.

Background Basic Properties of Integers pp. 515–516

Summations and Products pp. 10–14

Mathematical Induction pp. 18–21

Lecture 1 The Well-Ordering Property p. 6

Divisibility pp. 31–32

Lectures 2, 3 Prime numbers pp. 65–67

Prime Number Theorem pp. 68–72

Long runs of composites p. 74

Greatest common divisors pp. 80–81

The Euclidean algorithm pp. 86–88

Lecture 4 The fundamental theorem of arithmetic pp. 97–100

Lecture 5 Infinitely many primes of the form 4n + 3 p. 102

Fermat factorisation pp. 112–113

Lecture 6 Linear Diophantine equations pp. 119–123


Lecture 7 Congruences pp. 127–133

Lecture 8 Linear Congruences pp. 139–141

Lecture 9 Linear Congruences (continued) pp. 139–141

Lecture 10 The Chinese Remainder Theorem pp. 143–146

Lecture 11 The Chinese Remainder Theorem (continued) pp. 143–146

The Calendar pp. 179–182

Round-Robin tournaments pp. 184–185


8

Lecture 12 Wilson’s theorem pp. 197–198

Fermat’s little theorem pp. 199–200

Lecture 13 Euler’s theorem pp. 215–218

Lecture 14 Multiplicative functions pp. 221–225

Lecture 15 Sum and number of divisors pp. 232–235

(Theorem 7.8 on p.233 can be omitted)

Modular Exponentiation pp. 133–134

Lectures 16, 17 Public-key Cryptography pp. 285–290


Lectures 18, 19 Primitive roots pp. 307–312

Lectures 20, 21 Primitive roots for primes pp. 315–317

Lecture 22 Index Arithmetic pp. 329–332


Lecture 23 Quadratic Residues pp. 375–377

Lecture 24 Quadratic Residues and the Legendre Symbol pp. 378–382

(Omit Gauss’ Lemma. Only the statement of

Theorem 11.6 is needed and not the proof.

Lecture 25 Quadratic reciprocity pp. 392–394



Lecture 26 The Jacobi symbol pp. 404–407



Lecture 27 Pseudoprimes pp. 205–206

(Omit Theorem 6.6 on p. 206)

Euler Pseudoprimes pp. 412–413

(Omit Theorem 11.14 on p. 413.)


Lectures 28, 29 Pythagorean triples pp. 481–486

Lectures 30, 31 Some Diophantine equations pp. 487–494

Lecture 32 History of Fermat’s Last Theorem pp. 487–494


9

5 COURSE OUTLINE (using the third edition)

For those students who have the older third edition of Rosen.

TEXT BOOK REFERENCES TO ROSEN (3RD ED.)

We will assume that you are already familiar with the background material from

Chapter 1.

Background Basic Properties of Integers pp. 4-5

Summations and Products pp. 9-12

Mathematical Induction pp. 15-19

Lecture 1 The Well-Ordering Property p. 6

Divisibility pp. 36-38

Lectures 2, 3 Prime numbers pp. 64-66

Long runs of composites pp. 68-69

Greatest common divisors pp. 74-77

The Euclidean algorithm pp. 80-81

Lecture 4 The fundamental theorem of arithmetic pp. 90-94

Lecture 5 Infinitely many primes of the form 4n + 3 pp. 95-96

Fermat factorisation pp. 104-105

Lecture 6 Linear Diophantine equations pp. 112-115


Lecture 7 Congruences pp. 119-125

Lecture 8 Linear Congruences pp. 131-133

Lecture 9 Linear Congruences (continued) pp. 131-133

Lecture 10 The Chinese Remainder Theorem pp. 135-137

Lecture 11 The Chinese Remainder Theorem (continued) pp. 135-137

The Calendar pp. 166–170

Round-Robin tournaments pp. 171-172


10

Lecture 12 Wilson’s theorem pp. 185-186

(The converse of Wilson’s theorem on the

second half of p.186 is not required)

Fermat’s little theorem pp. 187-188

(The Pollard p− 1 method described at the

bottom of p.188 is not part of the course.)

Lecture 13 Euler’s theorem pp. 201-204

Lecture 14 Multiplicative functions pp. 207-212

Lecture 15 Sum and number of divisors pp. 217-220

(Theorem 6.8 on p.218 can be omitted)

Lectures 16, 17 Public-key Cryptography pp. 259-264


Lectures 18, 19 Primitive roots pp. 278-282

Lectures 20, 21 Primitive roots for primes pp. 285-287

Lecture 22 Index Arithmetic pp. 298-301

(Omit Theorem 8.17 onwards)


Lecture 23 Quadratic Residues pp. 331-333

Lecture 24 Quadratic Residues and the Legendre Symbol pp. 333-339

(Omit Gauss’ Lemma. Only the statement of

Theorem 9.5 is needed and not the proof.

Omit the material on pp. 340-341)

Lecture 25 Quadratic reciprocity pp. 348-350



Lecture 26 The Jacobi symbol pp. 357-361



Lecture 27 Pseudoprimes pp. 192-193

(Omit Theorem 5.6 on p. 193)

Euler Pseudoprimes pp. 367-369

(Omit Theorem 9.12 on p. 369.)


Lectures 28, 29 Pythagorean triples pp. 436-440

Lectures 30, 31 Some Diophantine equations pp. 442-445

Lecture 32 History of Fermat’s Last Theorem

(printed notes are supplied separately)


11

6 ASSIGNMENTS

ASSIGNMENT 1

1. The degree deg p of a polynomial

p = anxn + an−1x

n−1 + · · · + a0

with an 6= 0 is by definition the integer n. We define the binary relation

p ≺ q iff deg p < deg q.

(a) Verify that the axioms of partial ordering O1, O2, O3 are satisfied for the

above ordering relation in the set of all polynomials.

(b) Give an an example that shows that O4 does not hold.

(c) Show that the WOP holds for any set S that contains a non-zero poly-

nomial. (Notice that the zero-polynomial p ≡ 0 has no degree!) Hint.

Consider the set D of all degrees of the polynomials of S and apply WOP

to D.

2. Proof that the sum of the five consecutive non-negative integers is divisible by

5.

3. (a) Explain the meaning of d | a.

(b) If d | a and d | b, prove that d | (a + b).

(c) More generally, if d | a and d | b, prove that d | (xa + yb) for any integers

x and y.

4. Let d = gcd(a, b) = (a, b) and suppose that x and y are integers such that

xa + yb = m.

(a) Explain the meaning of (a, b).

(b) What is the relationship between d and m?

(c) If m = 1, show that d = 1.

(d) Suppose that xa + yb = 5. What are the possible values of (a, b)?

(Note: (c) is a very useful special case. When m > 1, we cannot conclude

that m = d.)

5. Find the gcd of 286 and 390 and express it in the form 286x + 390y.

6. Find the prime factorisations of each of the following integers.

(a) 111, (b) 2222220.

12

7. If (a, b) = 1 , a | c and b | c , show that ab | c. Give an example to show that

the result need not be true when (a, b) 6= 1.

8. Show that if a, b, and c are integers with (a, b) = (a, c) = 1, then (a, bc) = 1.

9. A student returning from Europe changes her Polish z lotys and Czech crowns

into Australian money. If she receives $7.90 and has received 44c for each

Polish z lotys and 7c for each Czech crown, what amounts of each type of

currency did she exchange, given that she started with at least 10 of each?

(Set the problem up as a linear Diophantine equation and use the methods of

this section).

10. Use Fermat’s method to factorise 17399.

13

ASSIGNMENT 2

1. Solve the following linear congruences (you may if you wish, find some solutions

by inspection):

(a) 3x ≡ 12 (mod 14), (b) 3x ≡ 5 (mod 7),

(c) 4x ≡ 4 (mod 6), (d) 4x ≡ 7 (mod 12).

2. Solve 286x ≡ 52 (mod 390).

(Hint: You may find the calculation you did in Assignment 1 Question 5

useful).

3. Use congruences to verify the divisibility statement 13 | (15529 + 1).

4. Solve x3 + 1 ≡ 0 (mod 3) using any method you wish.

5. Show that if a is an odd integer, then a2 ≡ 1 (mod 8).

6. Find an inverse modulo 19 of each of the following integers.

(a) 4, (b) 6, (c) 10.

7. In the ISBN of a book the fourth digit is unreadable, so the number is 3-76?3-

5197-7. Recover the missing digit knowing that∑10

k=1 kxk ≡ 0 (mod 11)

where xk is the k-th digit of the ISBN.

8. Set up a round-robin tournament schedule for (a) 7 teams, (b) 8 teams.

9. Assume {r1, . . . , rn} is a complete system modulo n. Show that, for any integer

a, the set {r1 + a, . . . , rn + a} is also a complete system.

10. Prove that a number n is divisible by 7 if and only if n′ − 2n0 is divisible by

7 where n0 is the last digit and n′ is the number obtained from n by deleting

the last digit. (Hint: Prove that −2n ≡ n′ − 2n0 mod 7).

Challenge problem1 If (m, n) = 1 and m ≥ 3, n ≥ 3, show that the congruence

x2 ≡ 1 (mod mn)

has solutions other than x ≡ ±1 (mod mn). (Hint. Prove that there exists x with

x ≡ 1 (mod m) and x ≡ −1 (mod n). Show that such x 6≡ ±1 (mod mn).)

1The challenge problems are harder and do not count for assessment

14

ASSIGNMENT 3

1. Illustrate the proof of Wilson’s Theorem with p = 13.

2. Using Fermat’s Little Theorem, find the remainder when 2100,000 is divided by

13.

3. Using Euler’s Theorem, find the remainder when 5100,000 is divided by 18.

4. Show that if n is odd and 3 does not divide n, then n2 ≡ 1 (mod 24).

(Hint: It may help to note that 24 = 3 × 8.)

5. Evaluate φ(n), τ(n) and σ(n) for each n such that 25 ≤ n ≤ 30.

6. Prove that if n is odd, then τ(n) ≡ σ(n) (mod 2).

7. Find the primes p and q if n = pq = 14647 and φ(n) = 14400.

8. What is the ciphertext that is produced when the RSA cipher with key e = 3,

n = 3763 is used to encipher the message GO?

9. The next meeting of cryptographers will be held in the town of 1739 0923. It

is known that the cipher-text in this message was produced using the RSA

cipher key e = 1997, n = 2669. Where will the meeting be held?

10. One of the oldest recorded cipher systems is the Caesar cipher, obtained by

using a cyclic shift of the letters of the alphabet. It is rather primitive com-

pared to the RSA system, and is easily broken. The following message was

sent using this code. Decipher the message, given that e is the most common

letter in English.

UIF DIFFTF JT HPPE

Challenge problem

(a) If n is odd and (n, 5) = 1, show that 5 | (n4 + 4n).

(b) For which positive integers n is n4 + 4n prime?

Hint for (b): Deal with the case n even first. For n odd, factorise n4 + 4n as a

product of two terms by completing squares.

15

ASSIGNMENT 4

1. (a) Find the order of the integers 2, 3 and 5 modulo 13.

(b) Find the order of the integers 2, 3 and 5 modulo19.

2. Find a primitive root for each of 13 and 19.

3. Find the order of the integers 1, 3, 7, and 9 modulo 10. Which of these numbers

are primitive roots modulo 10?

4. Show that if a is an integer relatively prime to the positive integer m and

ordma = st, then ordmat = s.

5. How many primitive roots are there for the prime 107?

6. Given that 5 is a primitive root for 23, express all the other primitive roots of

23 as powers of 5.

7. Write out a table of indices modulo 23 with respect to the primitive root 5.

8. (a) Find all the solutions of the congruence 3x5 ≡ 16 (mod 23).

(b) Find all the solutions of the congruence 13x ≡ 3 (mod 23).

9. Show that the odd prime divisors of the integer n2 + 1 are of the form 4k + 1.

(Hint: If p is an odd prime which divides n2 + 1, show that n has order 4

modulo p).

10. Show that if p is a prime and p ≡ 1 (mod 4), there is an integer x such that

x2 ≡ −1 (mod p). (Hint: Use theory to show that there is an integer x of

order 4 modulo p.)

16

ASSIGNMENT 5

1. Find all the quadratic residues and non-residues of 19.

2. Let p be prime and a a quadratic residue of p. Show that if p ≡ 1 (mod 4),

then −a is also a quadratic residue of p, while if p ≡ 3 (mod 4), then −a is a

quadratic nonresidue of p.

3. Evaluate each of the following Legendre symbols:

(a)

(19

23

), (b)

(17

79

), (c)

(18

101

).

4. Show that if p is an odd prime, then(−3

p

)= 1 if p ≡ 1 (mod 3),(

−3

p

)= −1 if p ≡ −1 (mod 3).

5. Evaluate the Jacobi symbol (22

35

)6. Show that the integer 644 is an Euler pseudoprime to the base 2.

7. Show that if n is an Euler pseudoprime to the bases a and b, then n is an Euler

pseudoprime to the base ab.

17

ASSIGNMENT 6

1. Find all primitive Pythagorean triples, x, y, z with z ≤ 13.

2. Find all Pythagorean triples x, y, z, not necessarily primitive, in which z = 50.

3. Let x, y, z be a primitive Pythagorean triple (with y even). Show

(i) z ≡ 1 (mod 4).

(ii) y is divisible by 4.

(iii) Exactly one of x, y is divisible by 3.

(iv) Exactly one of x, y, z is divisible by 5.

Deduce that xyz is divisible by 60.

4. Find a Pythagorean triple x, y, z (with y even) which cannot be written in the

form

x = m2 − n2,

y = 2mn,

z = m2 + n2,

with m, n integers. (Such an example will of course have to be non-primitive.)

5. Find all solutions in positive integers of the Diophantine equation

x2 + 2y2 = z2.

6. Let p be prime. Using Fermat’s little theorem, show that

(a) if xp−1 + yp−1 = zp−1, then p | xyz.

(b) if xp + yp = zp, then p | (x + y − z).

7. Show that the Diophantine equation x4 − y4 = z2 has no solutions in nonzero

integers. (Hint: Try the same method as used on x4 + y4 = z2. The problem

is hard.)

18

7 NUMBERS: A NUMBER THEORETIC CAL-

CULATOR

BACKGROUND

During our course we have seen that a calculator can be useful in some situations

involving division by large numbers. However, the ordinary hand held scientific

calculator is designed for use in calculus and statistics related areas, and is not ideal

for use in a number theory course. Rather than designing and marketing special

number theory calculators, manufacturers instead sell software which can run on

most computers, and which allow the user to do number theory calculations in

much the same way that a calculator operates.

If you have access to one of the large packages such as Maple or Mathematica,

you can use it to perform many of the calculations which arise in number theory.

(These packages can be used for many other things as well.) An alternative is to use

a non-commercial package such as NUMBERS, which was written by Ivo Duntsch of

the University of Osnabruck in Germany. Another alternative, if you have done some

computer programming, is to write your own programs for doing the calculations.

The rest of this section is devoted to describing how NUMBERS works.

GETTING STARTED

(Internals) The program NUMBERS is available in the Microcomputer Labora-

tories. To start the program, log on using the Windows Operating System. Open

the Maths folder, double click on the NUMBERS icon, then press any key to get

into the main menu. Alternatively, you can get into MS-DOS mode, type the word

numbers, and press the Enter or the Return key followed by any key to get into the

main menu.

(Externals) You can download the program from the unit web-site. You will

need access to a computer with windows, or DOS, or a DOS emulator, such as

DOSBox, in linux.

The program is menu driven and works essentially like a hand-held calculator.

You find the operation you want, and when prompted, you input the appropriate

numbers. NUMBERS then returns its answer. You may wish to make some se-

lections from the menus and try the program out yourself. Alternatively, you can

work through the following short practical session. About half an hour should be

enough time (provided you can get started!) to work through many of the following

problems, and to get a feel for how the system works. Please feel free to contact me

if you run into trouble.

The program is not case sensitive, so that you can use either lower case or upper

19

case letters when entering responses.

It is entirely optional whether you wish to see what this program can do. With

rare exceptions you should not use this program as an alternative to doing assign-

ment questions yourself using pencil and paper.

Nevertheless, I believe that there are some benefits in doing the following ex-

ercises, and similar ones which you should construct yourself. First of all you can

recall the exact meaning of the definitions that are involved in the exercises. You

can also think about what procedure you might use yourself if you only had pencil

and paper but a smaller problem. Finally, if you are interested in applications such

as the one to cryptography, the use of a computer package is essential to allow you

to handle realistically sized problems.

If you have never used a computer before, it is advisable to get help from someone

in order to start the program. NUMBERS is probably one of the easiest computer

applications to use once you get started.

EXERCISES

These exercises are based on the material in the early part of the unit.

(i) Arithmetic with Large Numbers. The following symbols are used for the

arithmetic operations.

+ Addition

- Subtraction

* Multiplication

/ Division with remainder

^ Powers

Exercise. Find 1357 and 79 + 1357.

Select 〈1〉 (Arithmetic) from the main menu. To do this you can just press the

1 key, or alternatively, use the arrow keys to highlight the selection you want, and

then press Enter. (Some keyboards have a key marked Return instead of Enter.)

In response to the prompt (Y/N) type N.

In response to the prompt 〈Enter the operation〉 type ^.

In response to the prompt 〈Enter the first operand〉 type 13 and press Enter.

In response to the prompt 〈Enter the second operand〉 type 57 and press Enter.

The result is

3124432031290254610011894949223517352998211575328796815860858733.

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Store this number in the first memory by pressing the F1 key. (The other keys

F2 to F6 can also be used.) Note how the display pauses to give you the opportunity

of saving the result. If you do not wish to save the result of your calculation, you

press any key to continue. Note how there is information given at the bottom of

the screen on what choices you can currently make. If you wish to get back to the

menus at any time, press the Esc key.

Now choose the + operation and enter 79 for the first operand. To enter the

second operand, recall the contents of the first memory by holding down the Shift

key and pressing F1. The result is

3124432031290254610011894949223517352998211575328796815860858812.

In the same way you can do other calculations with large numbers. Notice that

exact answers are given. If you try the same problems on an ordinary calculator,

the results are rounded off and given in scientific notation.

(ii) The Division Algorithm.

Exercise. Find the quotient and remainder when 1674823 is divided by 23555.

Choose the / operation. (If you are continuing on from the previous exercise,

this can be done immediately. Otherwise, get into the main menu and choose 〈1〉followed by the N option as above.) Enter 1674823 for the first operand and 23555 for

the second operand. The quotient 71 is displayed and you are given the opportunity

to save the number. Press any key and you then get the remainder 2418.

(iii) Greatest Common Divisor.

Exercise. Find gcd(225, 85).

Begin as in the above two examples. When prompted for the operation, type

G. For the first operand, enter 225 and for the second, enter 85. The result 5 is

displayed. (If you want to find numbers x and y such that 5 = 225x + 85y, treat

the problem as a linear Diophantine equation and consult the later example.)

(iv) Find the next prime.

Exercise. Find the smallest 7-digit prime.

Return to the main menu (use the Esc key if you are continuing on from the

previous exercise) and select the option 〈3〉 (Divisors.) From the next menu, choose

〈2〉 (Find next prime.) In response to the prompt, enter the number 999999. The

result is 1000003.

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(iv) Prime Factorisation.

Exercise. Factorise 5555566.

Get back to the main menu and press 3 to get the option called Divisors, and

then 3 again to get the option Factor an Integer. You are prompted to enter a

number. Type 5555566 and then press Return or Enter. The program states that

5555566 = 2 * 17 * 53 * 3083,

where each of 2, 17, 53 and 3083 are prime, and * means multiplication. (The

meanings of d, σ, s, Φ will be given later in our course.)

Exercise. Factorise 246913582444449753086467.

Proceed as above and enter this number in response to the prompt. The answer

is returned as

246913582444449753086467 = 1111111121 * 222222222222227.

This time the factorisation takes a little longer to do. The factorisation is hardest

to do (for numbers of a given length), when there are no small primes.

Exercise. Factorise

1252745380899336425989575235479711977423421689600728324418130238

587180844451464373483127279166578229248.

This 103-digit number factorises (in less than a second) as

1252745380899336425989575235479711977423421689600728324418130238

587180844451464373483127279166578229248

= 267 * 3136 * 11 * 37 * 79 * 139 * 73589 * 333667.

(Instead of typing in this number, you can produce it in the same way as I did.

As in the first exercise, calculate 1867 * 987654321123456789 and store it in one of

the memories. Then get back to the factorisation option and when prompted for n,

recall the contents of the memory.)

NUMBERS has a list of the first 20 primes, and it checks to see if any of these

are factors. If yes, it factors out that prime and works with the smaller number

remaining.

After that, it uses the Pollard-Brent algorithm. This algorithm (which is not

in our course material), is named after J.M. Pollard (the author of the original

Pollard rho method - see p.156 of Rosen), and Richard Brent, formerly Professor of

Computing Science at the Australian National University.

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Exercise. Enter a number of your own choice of about 30 to 40 digits, and see

how long NUMBERS takes to factorise it.

(v) Linear Diophantine Equation.

Exercise. Find a solution of 101x + 999y = 587.

NUMBERS uses the notation m0 x + m1 y = n. From the main menu, choose

the Divisors option and then the Solve a Linear Equation option. Type 2 for the

number of variables. Enter 101 in response to the prompt for m0. Enter 999 in

response to the prompt for m1. Enter 587 in response to the prompt for n. The

result is

267085 * 101 − 27002 * 999 = 587.

Once you have one solution, you can use theory to write down the general solution.

THE MANUAL

There is a manual for NUMBERS in electronic form on the disc. You type manual

to see it. A printed copy is available from me on request.

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