Elementary Number Theory

Unit I: Review of Arithmetic in $\mathbb{Z}$: Divisibility, gcd and lcm, Division algorithm, Euclidean algorithm. Prime numbers, Fundamental theorem of Arithmetic, Goldberg conjecture, Euclid’s and Euler’s proof of infinitude of primes, twine prime conjecture (statement only).

Unit II: Modular Arithmetic: Congruence relations - linear and polynomial, Fermat’s little theorem, Euler’s theorem, Wilson’s theorem, Chinese Remainder Theorem, Solutions of $X^{n}=1$

Unit III: Arithmetic functions: Definition and examples of arithmetic functions like Mobius functions, Euler’s $\phi$-function, Divisor function, the prime counting function $\pi(x)$. Dirichlet multiplication and Mobius inversion formula, Multiplicative functions, Dirichlet series and Euler product.

Unit IV: Quadratic Residue and Diophantine Equations: Primitive roots, index arithmetic, structure of unit group of $\mathbb{Z}/n\mathbb{Z}$. Quadratic residue, Legendre and Jacobi symbols, Quadratic Reciprocity law (statement only); Sums of two squares and four squares. Linear Diophantine equations, Pythagorean triples, Fermat’s last theorem; Continued fractions and rational approximations.