MTH 609

Division Algorithm, The greatest common divisor, Euclidean Algorithm and Diophantine Equations; Prime numbers and Fundamental Theorem of arithmetic; The theory of congruences; Linear congruences and The Chinese Remainder Theorem; Special congruences: Fermat's little theorem, Wilson's theorem; Euler's Phi-function and Euler's generalization of Fermat's little theorem; Applications: RSA cryptosystem; Legendre's symbol and its properties; Euler's criterion; Quadratic reciprocity law; Some nonlinear Diophantine equations; Representation of integers as sums of squares.