# Number Theory (MATH 38b, Spring 2010)

 Day & Time: Tuesday and Friday 3:10 — 4:30pm Location: Goldsmith, room to be announced Instructor: Joel Bellaiche Office Hours:Tuesday, 9:30-10:30am Teaching assistant: John Bergdall Textbook: Joseph H. Silverman A friendly Introduction to number theory, third edition, Pearson Prentice Hall , 2006. ISBN 0-13-186137-9. Prerequisites: Introduction to proofs (Math 23b) and either Math 22a or permission of the instrutor (email me). Examinations: There will be two in class tests. The first will be on Tuesday, February 23 and the second on Tuesday, April 6. The final exam is a take-home exam

Assessment: The course grades will be computed as follows:
30% Homework
20% First test
20% Second test
30% Take-Home Final exam

Tests and Exams: There are no makeups for missed tests. You must take the final examination at the time scheduled by the university; no final exams will be given earlier.

Homework: There will be two kinds of homework:

• Exercises give you a chance to practice and refine your basic calculus skills. They will not be collected.
• Graded homework: There will be seven graded homework due at regularly spaced dates to be annouced in class. The lowest grade will be dropped. Each of the other six is worth 5% of the final garde.

Written work: We write to communicate. Please bear this in mind as you complete assignments and take exams. You must explain your work in order to obtain full credit; an assertion is not an answer. student or professor.

Help: Help is available if you have trouble with homework or lecture material. My office hours are a good place to start. Discussions with your classmates can also be very helpful and are strongly encouraged. You may also ask the TA.

Objectives: This class is an introduction to number theory, that is the study of properties of natural integers (0,1,2,3,...) and related kind of numbers, like relative integers (...,-2,-1,0,1,2) and rational numbers (like 2/3 or -11/23). We shall cover a wide range of topics from elementary ones (divisibility, prime numbers and unique factorization, congruences) to more advanced, like the quadratic reciprocity law or elliptic curves. A unifying theme will be "diophantine equations", that is equations in several variables for which we look for solutions that are integers or rational numbers rather that real or complex numbers. We will study in detail Pell's equations, cubic equations, and Fermat's equation.

Proofs will play an important role in this course. First I shall give a proof of most of the theorems we will study in class, except for a some very hard results that will mainly be mentioned for cultural purposes. Moreover you will be required to produce and write proofs in exercises, graded homeworks, and exams. In a sense, this course is a continuations of the course "proofs" math 23a. I hope this course will illustrate the fundamental double importance of proofs in mathematics: cetifying that a result is correct, but also provide a better uderstanding of that result and suggest generalizations and other questions.

A very good novel about number theory : if you need some time off academic work but wish to stay in the world of numebr theory, try the novel: Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis.

## Schedule of Lectures

 Class Topic Read Suggested Exercises Due Jan 19 Introduction. Divisibility, the Greatest Common Divisor, Euclidean algorithm. §1,5,6 5.1,5.4. Jan 23 Bezout's theorem. The Fundamental Theorem of Arithmetic. Infinity of primes. §6,7,12,13 6.1,6.2,6.4,7.1,7.2,7.3,7.4 Jan 26 Consequences of the Fundamental Theorem of Arithmetic. Infinity of primes. Congruences. §7,12,13,8 12.5,13.2,13.3,13.4 Jan 29 Congruences. Little theorem of Fermat §8,9 8.2,8.3,8.4,8.5,9.1,9.2,9.3,9.4 Problems set 1 and Solutions (by John Bergdall) Feb 2 Congruences. Euler's theorem and Euler's phi function. §10,11 10.1,10.2,10.3,11.1,11.3,11.10,11.12,11.13 Feb 5 Euler's theorem and Euler's phi function §10,11 Feb 9 Power modulo m, roots, coding, Carmichael numbers §16,17,18 16.1,16.5,17.1,17.4,18.2,18.3 Problems set 2 and Solutions (by John Bergdall) Feb 12 Mersenne primes and perfect numbers §14,15 14.1,14.2,14.3,15.4,15.7 Feb 23 Midterm I §5,6,7,8,9,10,11,12,13,16,17,18 Solutions Feb 26 A formula involving Euler's phi function. Primitive roots §19,20 19.1,19.2 Mar 2 Primitive roots. §20,21 20.1,20.2,20.3,20.8,21.4 Mar 5 Squares modulo p §22,23 22.3,23.1 Problems set 3 Solutions (by John Bergdall) March 9 Squares modulo p, quadratic reciprocity, Dirichlet theorem on prime numbers §23,24,12 23.1,23.2,24.1,24.2,24.3,12.6 Mar 12 Using quadratic reciprocity. Rational numbers and diophantine approximation §30;35 Mar 16 Diophantine approximation; algebraic and transcendental numbers §30,35 Problems set 4 Solutions (by John Bergdall) Mar 19 Liouville's theorem. §35 Mar 23 Diophantine equations. Pythagorean triples. §2,3,4 Mar 26 Pythagorean triples with the geometric method. Remarks on Fermat Last Theorem. Square-Triangular numbers §4, 29 4.1,4.2; 29.1,29.3 Problems set 5 Solutions (by John Bergdall) April 9 Midterm 2 Sections 1 to 25 excepted 13 and 19. sections 29, 31, 35/ Training questions (with solutions) Extra Credit April 13 Pell's equation with D=2 and Square triangular number. §29,30 Apr 16 Pell's equation in general and diophantine approximation §29,32 Apr 20 Pell's equation. Sum of two squares §32,26,27 Apr 23 >Binomial cofficients and Pascal triangle §36 Problems set 6 Apr 27 Sum of two squares. Gaussian integers; §26,27;34 Apr 30 Gaussian Integer and unique factorization. Elliptic curves §34,43 May 4 (Last day of class) Elliptic curves §43,44,45,47,48 Extra Credit (solution of the first part; second part) May 7 (No class) Elliptic curves Problems set 7 May 10 (Noon) Take home final exam available here (and on your email) 1 to 12, 14 to 18, 20 to 27, 29 to 36, 43 to 48 May 12 (Noon) Due date for final exam.