Number Theory
This page is dedicated to Shaheed Bhai Mohar Singh Ji.
Photo of classroom activity:
Largest known prime at present is the Mersenne prime
Mp = 2 77 232 917– 1
This number was found in December 2017 and it has 23,249,425 digits.
The largest known perfect number PN is
PN = (2 77 232 917– 1)X Mp
Chapter 1 A Survey of Divisibility
Section 1.1
Section 1.1.2 Linear Combination Pages 6-7
Section 1.1.3 GCD
Section 1.2 Division Algorithm pages 11-18
Section 1.3 pages 18-24
Section 1.3 Euclidean Algorithm pages 24-29
Section 1.4 Diophantine Equations pages 30 to 36
Section 1.4 Diophantine Equations pages 36 to 41
Exercise 1.4 question 3(a) page 41
Exercise 1.4 question 3(a) page 41 finished off
Chapter 2 Primes and factorization
Section 2.1 Importance of Primes pages 45 to 47
Section 2.1 Fundamental Theorem of Arithmetic pages 47 to 53
Section 2.2 Ceiling and floor function pages 54 to 57
Section 2.2 Testing for Compositeness pages 57 to 61
Section 2.2 Primes pages 61 to 63
Section 2.3 Unsolved Problems in Number Theory pages 64 to 67
Section 2.3 Distribution of Primes pages 67 to 70
Section 2.3 Primes of 4n+1; 4n+3 pages 71 to 73
Section 2.3 Primes in an A.P. pages 73 to 76
Chapter 3 Theory of Modular Arithmetic
Section 3.1 Introduction to congruences pages 91-93
Section 3.1 Intro to congruences pages 94-98
Section 3.1 Intro to congruences pages 98-103
Section 3.1 Intro to congruences pages 103-4
Section 3.1 Intro to congruences pages 104-5
Section 3.1 Intro to congruences pages 106-8
The notes for this section are here:
Exercises 3.1 questions 9 and 10
Notes are here Exercises 3.1.pdf
Section 3.2 Congruent Properties of Multiplication pages 111 to 113
Section 3.2 Congruent Properties of multiplication pages 113-14
Section 3.2 Congruent Properties of multiplication pages 115-17
Section 3.3 Solving Linear Congruences pages 118 to 120
Section 3.3 Solving Linear Congruences pages 120 to 125
Section 3.3 Example 3.17 pages 125 to 126
Section 3.3 Inverse in modular arithmetic pages 126 to 128
Notes on the above are here Section 3.3
Section 3.4 Chinese Remainder Theorem pages 130 to 132
Section 3.4 Proof of the Chinese Remainder Theorem pages 133 to 136
Section 3.4.3 Applying the Chinese Remainder Theorem pages 136 to 138
Section 3.4.3 Example 3.25 Pages 138 to 140)
Notes on the above are here Section 3.4
Section 3.5 Factorization pages 141 to 144
Section 3.5 Factorization page 144 to 146
Section 3.5 Factorization by using modular arithmetic pages 146 to 147
Section 3.5 Factorization by using modular arithmetic pages 147 to 149
Here are the notes related to the above recordings:
Chapter 4 A Survey of Modular Arithmetic with Prime Moduli
Section 4.1 Proof of Fermat’s Little Theorem pages 153 to 156
Section 4.1.3 Applying Fermat’s Little Theorem pages 156 to 158
Applying Fermat’s Little Theorem pages 158-59
Section 4.1 Corollary to Fermat’s Little Theorem pages 159 to 162
Notes for these are here
Section 4.2 Wilson’s Theorem pages 163-65
Section 4.2 Proof of Wilson’s Theorem
Section 4.2.3 Converse of Wilson’s Theorem pages 168-69
Here are the notes Section 4.2.pdf
Section 4.3 Composite Integers pages 171-73
Section 4.3.2 Pseudoprimes pages 173-74
Section 4.3.3 Factorizing integers pages 175-77
Notes on section 4.3 are here Section 4.3.pdf
Section 4.4 Mersenne Numbers page 182-84
Section 4.4.2 Factorizing Mersenne numbers page 184-86
Section 4.4.2 Factorizing Mersenne numbers page 186-88
Section 4.4.3 Germain primes page 188-190
Section 4.4.3 Germain primes pages 191-93
The notes are attached for this section are here Section 4.4.pdf
Section 4.5 Perfect Numbers pages 195-98
Section 4.5 Perfect Numbers page 198-200
Section 4.5.4 The sigma function pages 200-205
Here are the notes for section 4.5:
Chapter 5 Euler’s Generalization of Fermat’s Theorem
Section 5.1.2 Euler’s totient function for primes pages 211-14
Section 5.1.3 Phi of prime powers pages 215-17
Section 5.1.4 Euler’s totient function of any natural number pages 217-19
Section 5.1.4 Euler’s totient function examples pages 219-22
Here are the notes for section 5.1 Section 5.1.pdf
Hardy (1877 – 1947) and Ramanujan (1887 – 1920).
A Video about Ramanujan and Hardy
A new film about Ramanujan and Hardy
The mathematicians patterns, like a painter’s or the poet’s, must be beautiful; the ideas, like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
G.H. Hardy in A Mathematicians Apology
William Thurston 1946 to 2012
I think most mathematicians love mathematics for mathematics’ sake. They really do like the feeling of being in an ivory tower. For the most part, they are motivated by applications. But I believe that, whatever their personal motivation is doing for mathematics, in most cases the mathematics they generate will ultimately have significant applications. The important thing is to do mathematics. But, of course, it’s important to have people thinking about applications too.
A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks. Gian-Carlo Rota – 1932 to 1999.
Notes on Number Theory
Corrections by Dr. Giovanna Scataglini Belghitar
Investigation in Cryptography by Shannon O’Brien is here.
Introductory Chapter
| Introductory chapter notes and exercises | Complete solutions to Introductory chapter | |
| Summary of results of Introductory Chapter | Appendix A |
Alan Turing 1912-1954
If you are an academic and would like complete solutions to the supplementary problems then send me an email by using your university email:
k.singh@herts.ac.uk
Chapter 1: Introduction to Number Theory
Chapter 2: Primes and Their Distribution
Test on Floor and Ceiling FunctionsTest on GCD and Prime Factorization - Section 2.2
Chapter 3: Modular Arithmetic
Test on modular arithmetic
Chapter 4: A Survey of Linear Congruences
Chapter 5: Euler’s Generalization of Fermat’s Theorem
Euler 1707 to 1783
Test on Euler's totient Function - Section 5.1
Chapter 6: Primitive Roots and Indices