Number Theory course - 1.1

We want to develop the most comprehensive, intuitive and exciting number theory course (for mathematics olympiad).

The first step toward that goal is to gather the important books.

1. Mathematical Circles by Fomin
2. Excursion in Mathematics by Bhaskaracharya Pratisthana
3. 104 Number Theory Problems by Titu Andreescu
4. Problem Solving Strategies by Arthur Engel
5. Elements of Number Theory by David Burton

Unfortunately none of the books are 'explicit' as far as the proof of relatively complicated theorems go. For example none of them have 'understandable' or 'intuitive' discussion on the proofs of Chinese Remainder Theorem or Wilson's Theorem. The point is not about 'dumbing down ' the matter. Feynman did not dumb down physics yet he managed to keep it interesting (Ref: Lectures on Physics).

CHEENTA Number Theory course work is split up into 14 sessions. Each session is of 180 minutes or three hours.

1. Session 1 - Mathematical Circles
2. Session 2 - Mathematical Circles
3. Session 3 - Mathematical Circles
4. Session 4 - Mathematical Circles
5. Session 5 - 104 Number Theory Problems
6. Session 6 - 104 Number Theory Problems
7. Session 7 - 104 Number Theory Problems
8. Session 8 - 104 Number Theory Problems
9. Session 9 - 104 Number Theory Problems
10. Session 10 - 104 Number Theory Problems
11. Session 11 - Problem Solving Strategies
12. Session 12 - Problem Solving Strategies
13. Session 13 - Problem Solving Strategies
14. Session 14 - Problem Solving Strategies

We want to solve about 200 (Problem Solving Strategies) + 200 (104 Number Theory Problems) + 150 problems (Mathematical Circles) = 550 problems inside classroom. Elementary Number Theory by Burton will be used as an additional theoretical companion whereas Excursion in Mathematics as an additional problem solving aid.

Please suggest improvements.

a nice problem from ISI 10+2

Compute I = \(\int_e^{e^4}\sqrt{log(x)}dx\) if it is given that \(\int _1^2 e^{t^2} dt = \alpha \)

I = \([x \sqrt{log(x)}]_e^{e^4} - \int_e^{e^4} x \frac{1}{2 \sqrt{log(x)}} \frac {1}{x} dx \)
= \([e^4 \sqrt {log_e e^4} - e \sqrt {log _e e}] - \frac{1}{2} \int_e^{e^4}\frac{1}{\sqrt{log(x)}} dx \)
= \(2 e^4 - e - \frac{1}{2} \int_e^{e^4}\frac{1}{\sqrt{log(x)}} dx \)

let log(x) = \(t^2\)

x =\(e^{t^2}\)

dx = 2t \(e^{t^2}\) dt

Thus I = \(2 e^4 - e - \frac{1}{2} \int_e^{e^4}\frac{1}{\sqrt{log(x)}} dx \)

=  \(2 e^4 - e - \frac{1}{2} \int _1^2 \frac {1}{t} 2 t e^{t^2} dt \)
=  \(2 e^4 - e -  \int _1^2 e^{t^2} dt \)
=  \(2 e^4 - e - \alpha \)

Number Theory Course Revamped

The topics that we learn and teach at CGK are:

• Math Olympiad topics
• Number Theory
• Geometry
• Combinatorics
• Functional Equation
• Inequalities
• Polynomials
• Trigonometry
• Complex Numbers
• High School Topics
• Calculus
• Algebra
• Coordinate Geometry
• Trigonometry
• Symbolic Logic
• Computer Science (C language)

We have stand alone courses on each sub topic. These stand-alone courses together make the entire mathematics program at Cheenta Ganit Kendra. Apart from the stand alone courses we have high intensity (inter-topic or topic-introducing) seminars which make the mathematics program worth experiencing.

NUMBER THEORY

As far as number theory is concerned, we have revamped the entire course. The course roughly constitutes.

Content

1. Divisibility 1 and 2 from Mathematical Circles Russian Experience - 4 problem sheets
2. Mathematical Induction and Binomial Theorem - 2 problem sheets
3. 104 Number Theory Problems - 4 problem sheets
4. 105 Diophantine Equation - 4 problem sheets
5. Number Theory section of Problem Solving Strategies by Arthur Engel - 4 problem sheets
6. Navigation through David Burton's Elementary Number Theory - 4 problem sheets

Power-seminars

1. Geometry inside congruency - 2 hours

Time-Line

Each problem sheet may take 5 hours of classroom work. The entire number theory course will be accomplished in 100 hours.