# Advanced Number Theory

**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.

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