Number Theory CST 292 KTU IV Semester Honors Course Notes Dr Binu V P 9847390760

About me

Syllabus

Scheme

Teaching Plan

Model Question Paper

University Question Papers

Assignment-1

Module-1 

Introduction

Well ordering principle

Group Ring Fields

Group Ring Fields in detail

Divisibility

Modular Arithmetic

GCD- Euclidean Algorithm

Bezout's Identity

Extended Euclidean Algorithm

LCM-Least Common Multiple

Linear Diophantine Equations

Modular Division

Module-II

Prime Numbers and Prime-Power Factorisation

Fermat and Mersenne Prime

Primality Testing and Factorization

Miller Rabin Primality Testing Algorithm

Fermat  Factorization

Congruence

Modular Exponentiation

Linear Congruences

Simultaneous Linear Congruences- Chinese Remainder Theorem

Fermat's Little Theorem and Euler's Theorem

Wilson's Theorem

Module-III

Congruence with Prime modulus

Congruence with Prime power modulus

Pseudoprime and Carmichael Numbers

Euler's Totient Function

Cryptography

The Group of Units - Primitive Roots

Module-IV

Quadratic Residues-Legendre and Jacobi Symbols

Arithmetic Functions-Mobius Functions, Dirichlet Products

Module-V

Sum of Squares

Continued Fractions-Pell's Equation



Comments

Post a Comment

Popular posts from this blog

Sum of Squares

Image
In this our aim is to determine which integers can be expressed as the sum of given number of squares ie; which have the form $x_1^2+x_2^2+\cdots+x_k^2$, where each $x_i \in \mathbb{Z}$, for a given $k$. Definition: For each integer $k \ge 1$, let $S_k=\{n|n=x_1^2+x_2^2+\cdots+x_k^2$ for some $x_1,x_2,\ldots,x_k \in \mathbb{Z} \}$ , the set of all sum of $k$ squares. Example: $S_1=\{0,1,4,9,\ldots\}$ is the set of all squares. By inspection $S_2$, the set of sum of two squares contains 0,1,2,4,5 and 8 but not 3,6, or 7. Lemma: The set $S_2$ contains, the sum of two squares is closed under multiplication ie; if $s,t \in S_2$ , then $st \in S_2$. Proof:Let $s=a_1^2+b_1^2$ and $t=a_2^2+b_2^2$ be the elements of $S_2$, where $a_1,b_1,a_2,b_2 \in \mathbb{Z}$. Then the identity $$(a_1^2+b_1^2)(a_2^2+b_2^2)=(a_1a_2-b_1b_2)^2+(a_1b_2+b_1a_2)^2$$ This shows that $st \in S_2$ Example: We have $8=2^2+2^2$ and $10=3^2+1^2$ so $8.10=80=(2.3-2.1)^2+(2.1+2.3)^2=4^2+8^2$ Note: an equivalent ident...
Read more