#### Diophantine Equation and Analysis – An Introduction ( # Lectures = 2)

Basic Overview of Number Theory – What number Theory is all about? Why suddenly the concepts of Divisibility and Primes come into play?

Diophantine Equation and Diophantine Analysis – Brief History, Definition, Examples.

Diophantine Equation and Number Theoretic Techniques – The Modulo Technique, Fermat’s’ Infinite Descent, Lagrange’s Technique.

#### Pell’s Equation (# Lectures = 1)

Construction of a Pell’s Equation – When Square Numbers gives rise to Triangular Numbers? leads to the Pell’s Equation $$x^2 – 2y^2 = 1$$.

Square – Triangular Number Theorem – All the solutions to $$x^2 – 2y^2 = 1$$ can be got by a single initial solution $$(3,2)$$, using a certain algorithm which involves an advanced application of Fermat’s Infinite Descent, which Andrew Wiles actually generalized later using height functions to prove Fermat’s Last Theorem.

General Solution of a Pell’s Equation – General Solution of a Pell’s Equation can be got if we know the fundamental solution of the equation, for example, $$(3,2)$$ in case of $$x^2 – 2y^2 = 1$$

Does always initial solution exist? – Why initial solutions exist always, that is a topic after we learn Gaussian Integers.

#### Imaginary Number Theory Incomplete ( # Lectures = $$\geq 2$$

The starting point of Imaginary Number Theory – The equation $$x^3 = y^2 + 2$$, which was solved by Euler in a unique way, that gave rise to the idea of Gaussian Integers.

The similarity to Integers – Divisibility, Division Algorithm, Primes, Units. The Proof that Division Algorithm Works by Gauss’s Method.

Reference: A Friendly Introduction to Number Theory by J.H.Silverman.

## 4 thoughts on “Advanced Number Theory”

1. Cheatum says:

Uncovered this write-up engaged, added to delicious

1. Srijit says:

Thanks, please do keep visiting. 🙂

2. راهنما فروش says:

Ciao My brother suggested I might like this website. He was entirely right. This post actually made my day. You can not imagine just how much time I had spent for this info! thank you

1. Srijit Mukherjee says:

Thanks brother.

Go Top