### 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 **

**M**_{p} = 2 ^{77 232 917}^{ }*– *1

_{p}= 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 M**_{p}

^{77 232 917}

^{ }

*–*1)

_{p}

##### Video lecture on section 6(a)

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

Proof by induction notes | Exercise on Mathematical Induction | Complete Solutions to Exercise |

Further Mathematical Induction notes | Exercise on Further Induction | Complete Solutions to Exercise |

#### 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

#### Chapter 3: Modular Arithmetic

#### Chapter 4: A Survey of Linear Congruences

**Chapter 5: Eulerâ€™s Generalization of Fermatâ€™s Theorem**

#### Euler 1707 to 1783

Chapter 6: Primitive Roots and Indices

#### Chapter 7: Quadratic Residues

#### Chapter 15: Continued Fractions