Math 25
Number Theory
Last updated May 31, 2008 12:24:21 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.

Lectures Sections in Text Brief Description
9/22 1.2, 1.4, and parts of 3.1, 3.4 Induction, Divisibility, Fundamental Theorem of Arithmetic
9/24 3.1 Primes: Euclid's and Euler's Proofs of Infinitude, Chebyshev's Estimates
9/27 3.2, beginning of 3.4 GCDs, the Fundamental Theorem Revisited
9/29 3.3, remainder of 3.4 Uses of the Fundamental Theorem, the Euclidean Algorithm
10/1 3.6, 13.1 Some Diophantine Equations: Linear Equations, Pythagorean Triples
10/4 13.1, 13.2, 4.1 Pythagorean Triples (cont.), a Special Case of Fermat's Last Theorem, Intro to Congruences
10/6 4.1 Intro to Congruences (cont.)
10/8 4.2 Primes in Progressions, Linear Congruences
10/11 4.3 Chinese Remainder Theorem
10/13 Supplemental and 4.4 Reclusive Primes, Solving Polynomial Congruences
10/15 4.4 Hensel's Lemma
10/18 5.1, 5.2 Applications of Congruences: Divisibility Tests, The Perpetual Calendar
10/20 5.2, 5.5 The Perpetual Calendar (cont.), Check Digits
10/22 6.1 and Supplemental Wilson's Theorem, Formula for the nth Prime Number
10/25 4.6, 6.1 Fermat's Little Theorem, Pollard Rho and Pollard p-1 Factorization
10/27 6.2 Primality Tests, Pseudoprimes and Carmichael Numbers
10/28 (x-hour) 6.2, 6.3 Miller's Test and Rabin's Probabilistic Primality Test, Euler's Theorem
11/1 8.1, 8.2 Intro to Cryptography: Character and Block Ciphers
11/3 4.5, 8.2 Hill Ciphers and Systems of Linear Congruences
11/5 Supplemental A Little Bit of Cryptanalysis
11/8 8.2, 8.3 Cryptanalysis Continued, Stream Ciphers, Exponentiation Ciphers
11/10 8.4 Public Key Cryptography and RSA, Cryptographic Protocols and Applications
11/12 Supplemental Arithmetic Functions, Dirichlet Convolutions, Formal Dirichlet Series
11/15 7.1, 7.2 Euler's Phi-function, Sum and Number of Divisors
11/17 7.3, 7.4 Perfect Numbers, Mersenne Primes, Lucas-Lehmer Test, Mobius Inversion
11/19 Supplemental Average Orders of Multiplicative Functions, Counting Square-Free Integers, Farey Fractions
11/22 11.1 Quadratic Residues, the Legendre Symbol