Math 25
Elementary Number Theory

Last updated August 02, 2011 13:48:42 EDT

## Syllabus

The following is a tentative syllabus for the course. This page will be updated irregularly.
On the other hand, the weekly syllabus contained in the Homework Assignments page will always be accurate.

We shall cover chapters 1 - 8 of the text with some additional material on cryptography.

Lectures Sections in Text Brief Description
9/23 1.1, 1.2 Introduction, Division and Euclidean algoritms
9/25 1.2, 1.3 Bezout's identity, least common multiples
9/28 1.3, 1.4 LCMs, Linear Diophantine Equations
9/30 2.1, 2.2 Fundamental Theorem of Arithmetic, Distribution of Primes
10/2 2.2, 2.3 Distribution of primes, Fermat/Mersenne Primes
10/5 2.3, 3.1 Distribution of primes, modular arithmetic
10/7 3.2 Linear congruences
10/9 3.3 Chinese Remainder Theorem
10/12 3.1, 3.4 Polynomials and polynomial congruences mod d
10/14 4.1 Arithmetic os Z_p
10/16 4.2 Pseudoprimes and Carmichael numbers
10/19 4.2, classnotes Pseudoprimes and strong pseudoprimes
10/20   First Hour Exam
10/20 (x-hour)   In class part with take-home due tomorrow
10/21 classnotes strong pseudoprimes, Miller's test
10/23 5.1, 81. Euler's function
10/26 5.2, 8.1 Multiplicative functions, Euler's function
10/28 5.3, classnotes Crytpography, RSA
10/30 classnotes Public Key Cryptosystems, signatures, and authentication
11/2 6.1, 6.2 U_n and primitive roots modulo primes
11/4 6.3, 6.4, 6.5 primitive roots for composite moduli, indices
11/6 6.6, 7.1 indices, applications of primitive roots, quadratic residues
11/10   Second Hour Exam
11/10 (x-hour)   In class part with take-home due tomorrow
11/11 7.3 The Legendre symbol and properties
11/13 7.4 Quadratic Reciprocity, Fermat Numbers
11/16 7.4, 7.5 Mersenne Numbers, Pepin's Test, quadratic residues mod p and mod p^e
11/18 8.2-8.6, class notes Perfect, abundant, deficient numbers; Mobius Inversion
11/20 8.2 - 8.6 Mobius inversions, Dirichlet convolution
11/23 10.1, 10.3, 10.4 Sums of 2, 3 and 4 squares
11/25   Thanksgiving break: 11/25 - 11/29
11/30 10.2 Gaussian integers and sums of two squares
12/2 10.5 Quaternions and sums of four squares

T. R. Shemanske
Last updated August 02, 2011 13:48:42 EDT