This document is useful for current students. It contains teaching schedule for Number Theory 1.
Overview:
Number Theory 1 is an introductory module. It is useful for beginner math olympiad aspirants (preparing for AMC, AIME, ARML, Duke Math Meet etc.)
- Number systems
- Prime numbers
- Arithmetic and geometric sequences
- Mathematical Induction
- Divisibility techniques
- Arithmetic of remainders
- Modular Arithmetic and Gauss’s theory
- Equivalence Relations
- Mathematical games
Specifications
- Each session (day) is 2 hours long.
- It is followed by a homework assignment.
- Apart from regular theoretical work and problem-solving, each section consists of mathematical games
- Books:
- Challenges and Thrills of Pre-College Mathematics
- Mathematics can be Fun by Yakov Perelman
- Excursion Into Mathematics
- Mathematical Circles, Russian Experience by Fomin
Sessions
Session 1
- Formula for nth odd number and nth even number
- Sum of first n odd numbers and their visual treatment
- Arithmetic Progression
Session 2 and 3
- Arithmetic Progression’s description (nth number)
- Gauss’s method for summing arithmetic progression (rewriting a finite sum in reverse order).
- Sum of n terms of an arithmetic sequence
- Geometric sequence
- Sum of nth term of a geometric sequence
Session 4 and 5
- Mathematical induction
- Strong form of induction
Session 6 and 7
- Divisibility and prime numbers
- Fundamental Theorem of Arithmetic
Session 8
- Types of numbers
- Well ordering principles
- Irrationality of square root of 2 (and primes)
Session 9 and 10
- Modular Arithmetic – similarity and differences with equality
- Notion of equivalence relation
