Math 431	Spring 1998
Instructor: Abraham Goetz
Text:  A Friendly Introduction to Number Theory by J.H.Silverman, Prentice Hall 1997.
The book:  Fundamental Number Theory with Applications by Richard A. Mollin, CRC Press   
was places in the Math Library on the reserved books shelf.

I covered Chapters 1 to 27. The Gauss Lemma used in Chapter 23 (without the name an in a 
special case a = 2) was stated and proved in whole generality.

Syllabus

1.	Integers and their properties. Divisibility. The greatest common divisor of two 	numbers. The 
	division theorem. Euclid's algorithm. Representation of the gcd. of two numbers as their linear 
	combination with integer coefficients. Relatively prime numbers.
2.	Congruences modulo m..  Properties of congruences. Solutions of a congruence 
	ax º b  (mod n).The Chinese Remainder Theorem. Application of congruences to divisibility 
	tests: divisibility by 3 and 9. Divisibility by 11.
3.  Prime numbers. The prime factorization theorem. Twin primes. The set of prime numbers is 
	infinite. The prime number theorem. Little Fermat's Theorem. Euler's j-function (the totient 
	function) and the generalization of Little Fermat's Theorem.  Mersenne primes. The function 
	s(n). Perfect numbers and their relation to Mersenne primes. The Internet site for large primes 
	http://utm.edu/research/primes
4.   Computation of powers modulo n by the method of consecutive squares. Solving the
	congruence xm ºb   (mod n)  in the case, when j(n) is known and is relatively prime to the 
	exponent m.. RSA codes
5. Multiplicative functions. examples of multiplicative functions j(n), s(n). The sum  	
	F(n) = åd|nÊÊf(d), where, f (n) is a multiplicative function, is multiplicative. 
6. Powers modulo p  (p a prime). The exponent ep (a) of a number modulo p.  ep (a) | p - 1. 
	Primitive roots modulo p. Number of primitive roots. General case: powers modulo m. 
	Exponent em (a). em (a) | j(m). Primitive roots modulo m.
7.  Indices of a number modulo p, where p is a prime, for a given primitive root.Analogy with 
	logarithms. Using indices for computation.

8.	Quadratic residues and non-residues modulo m.  The number of quadratic residues between 1 
	and p - 1 modulo a prime p. The Legendre symbol ()nm. The Legendre symbol  ()nm is a 
	multiplicative function of n, when m is fixed. Euler's criterion.The Gauss lemma. The 
	quadratic reciprocity theorem: ()-1p and  ()2p with proof,  ()pq without proof.

9.  Representation of primes as sum of two squares. The Fermat descent procedure. The 
	representation of any number as a sum of two squares. Which numbers can be the 
	hypotenuse of a primitive Pythagorean triangle.

10. Nonsolvability of the equation x4 + y4 = z2 in nonzero integers. A few words about the 
	Wiles proof of the Last Fermat Theorem.

Comments
The textbook is very well written and accessible for the students to read. It has a very nice selection 
of more advance topics in addition to the basics. I consider it as the best text for an undergraduate 
course I encountered.
There were two tests during the semester, each consisting of a take-home part, which contained the 
problems requiring more computations, and a in-class part more theoretical.
The final was a take-home test asking for a written presentation of the teory of congruences 
covered in the course.