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