Introduction to Number Theory

Syllabus Details

• Unit I: (Review of arithmetic in $\mathbb{Z}$) Divisibility, gcd & lcm, Bezout’s identity, Division Algorithm and Euclidean Algorithm
• Unit II: (Modular Arithmetic): Congruence relation, Euler phi-function, Fermat’s little theorem, Euler’s theorem, Wilson’s theorem, CRT, primitive roots and structure of units modulo $n$. Solving equations $X^{n}=1$, Euclid’s and Euler’s proof of infinitude of primes
• Unit III: (Quadratic Reciprocity Law): Quadratic residue, Legendre & Jacobi symbols, Law of Quadratic Reciprocity (at least two proofs) with application in solving quadratic equations, Sums of Two Square theorem and Lagrange Four Square theorem, Primes in arithmetic progressions ($\text{mod } 8$)
• Unit IV: (Arithmetic Functions): Some examples of arithmetic functions, Möbius inversion formula, Big-$O$ and little-$o$ notations, Abel’s partial summation formula with applications, Prime Number Theorem (statement only) and Dirichlet’s theorem on primes in arithmetic progressions (statement only)
• Unit V: (Continued Fractions): Existence and uniqueness of simple continued fraction expansion, periodic continued fraction expansions, Equivalent numbers
• Unit VI: (Algebraic Numbers): Algebraic and transcendental numbers, proof of transcendence of $e$, rings of algebraic integers (quadratic number fields), Gaussian integers and prime factorizations

Course Learning Objectives

1. Learn modular arithmetic and solving linear Diophantine equations.
2. Understand Fermat’s/Euler’s theorem and applications to classify integers of special types (sums of two/four Squares).
3. Solve quadratic equations using Quadratic Reciprocity Law.
4. Learn Continued fractions and apply to solve some of Diophantine equations (Linear with two variables, Pell’s equations etc).
5. Understand Algebraic and Transcendental numbers with explicit examples, prime factorizations in $\mathbb{Z}[i]$.

Recommended Books

1. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 6th edition, Oxford University Press, 2008.
2. G. A. Jones and J. M. Jones, Elementary Number Theory, Springer-Verlag, London, 1998.
3. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM 84, Springer-Verlag, New York, 1990.
4. D. M. Burton, Elementary Number Theory, 7th Edition, McGraw-Hill, 2010.
5. T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, Heidelberg, 1976.