MA 405 Basic Algebra
Infinitude of primes, discussion of the Prime Number Theorem, infinitude of primes in specific arithmetic progressions, Dirichlet's theorem (without proof).
Arithmetic functions, Mobius inversion formula. Structure of units modulo n, Euler's phi function
Congruences, theorems of Fermat and Euler, Wilson's theorem, linear congruences, quadratic residues, law of quadratic reciprocity.
Binary quadratics forms, equivalence, reduction, Fermat's two square theorem, Lagrange's four square theorem.
Continued fractions, rational approximations, Liouville's theorem, discussion of Roth's theorem, transcendental numbers, transcendence of "e" and "pi".
Diophantine equations: Brahmagupta's equation (also known as Pell's equation), the Thue equation, Fermat's method of descent, discussion of the Mordell equation.
Optional Topics:
Discussion of Waring's problem.
Discussion of the Bhargava-Conway "fifteen theorem" for positive definite quadratic forms.
The RSA algorithm and public key encryption.
Primality testing, discussion of the Agrawal-Kayal-Saxena theorem.
Catalan's equation, discussion of the Gelfond-Schneider theorem, discussion of Baker's theorem.
W.W. Adams and L.J. Goldstein, Introduction to the Theory of Numbers, 3rd ed., Wiley Eastern, 1972.