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.

After this, the students told that they wanted to solve more problems than theory. I then shifted my focus to Problem Solving in the Level 3 class.

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