\noindent {\bf Math 431	Spring 2000 Syllabus}\smallskip
\noindent {\bf Instructor:} Matthew Dyer\smallskip
\noindent{\bf Text:}  A Friendly Introduction to Number Theory by J.H.Silverman,
Prentice Hall 1997.\smallskip

\noindent We covered the following material from Chapters 1 to 27 and 32 to
37:\smallskip 



1) Description of Pythagorean triples and primitive Pythagorean triples,
 Solutions of $x^2+y^2=1$ in rational numbers.
	
2) Integers. Divisibility. The greatest common divisor of two 
numbers. The 	division theorem. Euclidean  algorithm. Representation of the gcd of
two numbers as their linear 	combination with integer coefficients. Relatively prime
numbers. Solutions of $ax+by=c$ in integers.

3) 	Congruences modulo $m$.  Properties of
congruences. Solutions of a linear congruence 
	$ax \equiv  b  \pmod m$. The Chinese Remainder Theorem.

4) Prime numbers. Factorization and the fundamental theorem of arithmetic.
Infinitude of primes, of primes of  form $4k+3$.
Statement of prime number theorem, Dirichlet's theorem on primes in arithmetic
progression and conjectures on twin primes and primes of form
$n^2+1$. Mersenne primes. Euler's theorem on even perfect numbers.
Question of non-existence of odd perfect numbers.

5) Fermat's little theorem. Euler's $\phi$-function, formulae for evaluation of
$\phi$ (multiplicative nature of $\phi$ and its value on prime powers), Euler's
generalization of Fermats little theorem. Multiplicative functions, the function
$\sigma(n):=\sum_{d\mid n} d$,  theorem on multiplicative nature of 
$F(n)=\sum_{d\mid n} f(d)$ for a multiplicative function $f$,
formula $\sum_{d\mid n} \phi(d)=n$.

6) Calculations of powers modulo $m$ by successive squaring. Computing the $k$-th
root of $a$ modulo $m$ for a relatively prime to $m$ and $k$ relatively prime to
$\phi(m)$. RSA public key cryptosystem.

7) Exponents of integers $a$ modulo $p$ (prime). Number of solutions of $a^d\equiv
1\pmod p$. Existence of a primitive root modulo $p$. Indices 
 and their applications to computation of powers modulo $p$ and
solution of equations $ax^d\equiv b\pmod p$.

8) Quadratic residues  and non-residues modulo $p$. Characterization of QR and NR
by parity (evenness or oddness) of their indices. Number of QR and NR. The Legendre
symbol and its multiplicative nature (QR/ NR ``multiplication
table''). Euler's criterion   ($a^{(p-1)/2}\equiv \bigl({a\over p}\bigr)\pmod p$.
Gauss' lemma (for odd $p$, $\bigl({a\over p}\bigr)=(-1)^n$ where $n$ is the number of
negative integers amongst $[a1], [a2], \ldots, [a(p-1)/2]$ and $[j]$ denotes the
unique integer with $-(p-1)/2<[j]<(p-1)/2$ and $[j]\equiv j\pmod p$). 
Law of Quadratic Reciprocity (proof of formulae for values of
 $\bigl({-1\over p}\bigr)$,  $\bigl({2\over p}\bigr)$ and statement of 
 the relationship between  $\bigl({q\over p}\bigr)$ and
 $\bigl({p\over q}\bigr)$ for odd primes $p$, $q$). Existence of infinitely many
primes of the form $4k+1$.

9) Characterization of primes which are sums of two squares, Fermat's descent
proceedure. Statement of which integers are expressible as a sum of two squares
(proved only that integers whose ``square-free part'' is not divisible by a prime
$p\equiv 3\pmod m$ are a sum of two squares). Non-existence of positive integer
solutions of $x^4+y^4=z^2$ (Fermat's last theorem for exponent $4$). 


10)  Elliptic curves (over the rational numbers). Formula for discriminant.
Constructing new points on elliptic curves by taking line through two known points
and intersecting with the curve and statement this gives a group structure.
Statement of Mordell's theorem on finite genration of rational points. Torsion
collections of rational points on elliptic curves (finite subgroups of the group of
rational points), statements of Nagell-Lutz theorem and  Mazur's theorem on torsion
collections, and of  Siegel's theorem on finiteness of integral points. Points on
elliptic curves modulo
$p$. Definition and computation  of $p$-defects.
Constructing new points modulo $p$ by taking ``lines''  through two known points
modulo $p$ and intersecting with the curve and statement this gives a group structure
provided there is good reduction at $p$.  Implications of existence of a torsion
collection of
$t$ points  for patterns in $p$-defects modulo
$t+1$. Statement of Elkie's theorem on existence of infinitely many primes $p$
for which  $p$-defect is $0$ and  statement of Hasse's bound on
$p$-defects. Explanation of what is meant  a modular form and a ``modularity
pattern'' in the
$p$-defects of an elliptic curve. Statement of Taniyama-Shimura conjecture and 
description of overall structure of  proof of Fermat's last theorem.
(We illustrated the theory by computing with  specific elliptic
curves $E1:y^2=x^3+17$,  $E2:y^2=x^3+x$ and $E3:y^2=x^3-4x^2+16$.
We found many rational points on $E1$, proved $E2$ has only one rational point,
stated   a complete description of $p$-defects for $E2$, proved  the  
$p$-defects for  $E1$ are zero for $p\equiv 2\pmod 3$ and described the modularity
pattern for  $E_3$.) 
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