This course will strengthen each student's ability to use theory to solve concrete problems in number theory. Application of abstract algebra interms of number theory will enable student to apply these techniques to solve different problems in number theory.
Divisibility: Divisors, Bezeout’s identity, LCM, Linear Diophantine equations,
Prime Numbers: Prime numbers and prime-power factorizations, Distribution of primes, Primality-testing and factorization,
Congruences: Modular arithmetic, Linear congruences, An extension of chineses Remainder Theorem, The Arithmetic’s of Zp. Solving conruences mod (pe). Euler’s Function: Units, Euler’s function,
The Group of Units: The group Un, Primitive roots, The group Un, n is power of odd prime and n is power of 2,
Quadratic Residues: Quadratic congruences, The group of quadratic residues, The Legendre symbol, Quadratic reciprocity,
Arithmetic Functions: Definition and examples, perfect numbers, The Modius Inversion formula,
The Reimann Zeta Function: Random integers, Dirichlet series, Euler products, Sums of two Squares, The Gaussian integers, sums of three Squares, Sums of four Squares,
Fermat’s Last Theorem: The problem, Pythagorean Theorem, Pythagorean triples, The case n=4, Odd prime exponents.