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Math 312 Winter 2017 Term 1 -- Syllabus

Sections of the book to be covered (at least partially).

We will not necessarily follow this order.

1.3. Induction

1.5. Divisibility

2.1. Representations of integers.

3.1. Prime numbers

3.2. The distribution of primes (only the statement of the Prime Number Theorem)

3.3. Greatest common divisors

3.4. The Euclidean algorithm

3.5. The fundamental theorem of arithmetic

3.6. Fermat Factorization only.

3.7. Linear Diophantine equations

4.1. Introduction to congruences

4.2. Linear congruences

4.3. The Chinese Remainder Theorem

5.1. Divisibility tests

6.1. Wilson's Theorem and Fermat's Little Theorem

6.2. Pseudoprimes

6.3. Euler's Theorem

7.1. The Euler phi-function

7.2. The sum and number of divisors

7.3. Perfect numbers and sMersenne primes

8.1. Character ciphers

8.3. Exponentiation ciphers

8.4. Public key cryptography

9.1. The order of an integer and primitive roots.

9.2. Primitive roots for primes.

9.3. The existence of primitive roots.

9.4. Discrete logarithms and index arithmetic.

Depending on time availability we will also cover some of the following topics:

5.5. Check digits (ISBN code only)

8.6. Cryptographic protocols and applications (digital signatures only)

10.2 The ElGamal cryptosystem.

11.1 Quadratic residues and nonresidues

11.2 The Law of Quadratic reciprocity

11.1 The Jacobi Symbol

13.1 Pythagorean Triples

13.2 Fermat's Last Theorem (case n=4 only)