Euler’s Generalization of Fermat’s theorem. Euler’s theorem. Some Properties of the Phi-Function. Primitive Roots and Indices. The Order of an integer modulo n primitive roots for primes. Composite numbers having primitive roots. The Quadratic Reciprocity law. Legendre symbols and its properties. Quadratic congurences with composite moduli. Introduction to Cryptography. Numbers of Special Form. Certain Nonlinear Diophantine Equations. Fermat’s Last Theorem. Representation of integers as sums of squares. Fibonacci Numbers. Finite and infinite continued fractions Pell’s Equation.

Learn about the arithmetic of algebraic number fields, increase student interest and understanding and generate student interaction. learn to apply in Art, Coding Theory, Cryptology, Computer Science and other necessities of modern life. learn to prove theorems about integral bases, and about unique factorization into ideals., learn to calculate class numbers, and to use the theory to solve simple Diophantine equations.

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