Number Theory CST 292 KTU Honors IV Semester - Dr Binu V P

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 Func

 

 

NUMBER THEORY- SYLLABUS Module 1:Divisibility and Modular Arithmetic: Finite Fields – Groups, Rings and Fields. Divisibility - Divisibility and Division Algorithms, Well ordering Principle,Bezout’s Identity. Modular Arithmetic- Properties, Euclid's algorithm for the greatest common divisor, Extended Euclid’s Algorithm, Least Common multiple, Solving Linear Diophantine Equations, Modular Division. Module 2:Primes and Congruences: Prime Numbers-Prime Numbers andprime-powerfactorization, Fermat and Mersenne primes.,Primality testing and factorization. Congruences-Linear congruences, Simultaneous linear congruences, Chinese Remainder Theorem, Fermat’s little theorem, Wilson's theorem. Module 3:Congruences with a Prime-Power Modulus&Euler's Function: Congruences with a Prime-Power Modulus-Arithmetic modulo p, Pseudoprimes and Carmichael numbers, Solving congruences modulo prime powers. Euler's Function-Euler’s Totient function, Applications of Euler’s Totient function, T

Number Theory CST 292 Question Paper  July 2021 Number Theory CST 292 Question Paper June 2022 Number Theory CST 292 Question Paper  June 2023

Welcome to the fascinating world of number theory! Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers. It's one of the oldest branches of mathematics, dating back to ancient times, and it continues to be a vibrant and active field of research today. At its core, number theory focuses on the study of integers and their properties. Integers are whole numbers, including positive numbers, negative numbers, and zero. Number theorists explore various aspects of these integers, seeking patterns, relationships, and structures within the numerical realm. One of the central themes in number theory is prime numbers. Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. They play a fundamental role in number theory and are often considered the building blocks of the integers. The distribution and properties of prime numbers have intrigued mathematicians for centuries and continue to be a

 

Divisibility  We say that a nonzero $b$ divides $a$ if $a=qb$ for some $m$, where $a$,$b$ and $q$ are integers.That is $b$, divides $a$ if there is no remainder on division.The notation $b \mid a$ is commonly used to mean$b$ divides $a$.Also, if  $b|a$, we say that $b$ is a divisor of $a$. If $b$ does not divide $a$ we denote it by $b \nmid a$. For example, $6 \mid 36$  but $7 \nmid 36$. Note that $b \mid 0$ for any non-zero integer $b$ and $1\mid a$ for any integer $a$. When we write $b \mid a$, it is tacitly assumed that $b$ is a non-zero integer. We can easily deduce the following properties form the definition of divisibility itself Let $a, b, c ,d$ be any non-zero integers. 1.if $a|1$ , then $a = \pm 1$ 2. If $a | b$ and $b|c $  then  $a |c$. Proof: (1) Suppose $b = na$ and $c = mb$, where $n; m \in Z$. Then $c = m(na) = (mn)a$ and so $a | c$. 3. If $a | b$ and $c| d$ then $ac | bd.$ Proof: (2) Suppose $b = na$ and $d = mc$ where $n; m \in Z.$ Then $bd = (na)(mc) = (mn)(ac)$, i

  Groups : A  group  is a mathematical structure that consists of a set of elements along with an operation (usually called  addition  or  multiplication ). Key properties of a group: Closure: The result of the operation on any two elements in the group remains within the group. Associativity: The operation is associative (i.e., the order of performing the operation doesn’t matter). Identity element: There exists an element (often denoted as  e ) such that combining it with any other element doesn’t change the other element. Inverse element: Each element has an inverse (opposite) element within the group. Example: The integers under addition form a group. Rings : A  ring  is a more general structure than a group. It includes both  addition  and  multiplication  operations. Key properties of a ring: Closure under addition and multiplication. Associativity of both operations. Distributive property: Multiplication distributes over addition. Unlike groups, rings need not have multiplicativ

Groups, rings, and fields are the fundamental elements of a branch of mathematics known as abstract algebra, or modern algebra. In abstract algebra, we are concerned with sets on whose elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set. These operations are subject to specific rules, which define the nature of the set. By convention, the notation for the two principal classes of operations on set elements is usually the same as the notation for addition and multiplication on ordinary numbers. Let's break down the concepts of groups, rings, and fields in the context of number theory, and then we'll discuss the notion of group rings and fields. 1. Groups: In mathematics, a group is a set equipped with an operation that combines two elements to produce a third, satisfying four fundamental properties: Closure: The operation combines elements to produce another element in the set.

In mathematics, the well-ordering principle states that every non-empty set of positive integers contains a least element. In other words, the set of positive integers is well-ordered by its "natural" or "magnitude" order in which $x$ precedes $y$ if and only if  $y$ is either $x$ or the sum of $x$ and some positive integer (other orderings include the ordering $2,4,6$; and $1,3,5$). The phrase "well-ordering principle" is sometimes taken to be synonymous with the "well-ordering theorem". On other occasions it is understood to be the proposition that the set of integers $\{…,−2,−1,0,1,2,3,…\}$ contains a well-ordered subset, called the natural numbers, in which every nonempty subset contains a least element. Well-ordering Principle: If $S$ is a non-empty set of non-negative integers, then $S$ has a least element, i.e., there is an integer $c \in S$ such that $c \le x$ for all $ x \in S$. The principle of mathematical induction follows direct

Modular arithmetic is a fascinating branch of mathematics that deals with numbers “wrapping around” when they reach a certain value, known as the modulus. Here are the key concepts: Definition: In modular arithmetic, we work with integers and consider their remainders when divided by a fixed quantity (the modulus). Think of it like a clock: after reaching 12 hours, the clock “wraps around” to 1. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss The Modulus If $a$  is an integer and $n$ is a positive integer, we define $ a \: mod \: n $ to be the remainder when $a$ is divided by $n$ . The integer is called the modulus. Thus, for any integer , $$a = qn+r \quad  0 \le r \lt n \quad q=\lfloor a/n \rfloor$$ $$r= a - \lfloor a/n \rfloor .n $$ Example: $11 \: \mod \: 7 =4$ and $-11 \: \mod  \: 7=3$ Two integers $a$ and $b$ are  said to congruent modulo  $n$, if $a \: \mod \: n= b \: \mod \: n$.This is written as $$a \equiv b (\mod  n)$$ Example: $5 \equiv 16 (

Bézout’s identity, also known as Bézout’s lemma, is a fundamental theorem in elementary number theory. Named after Étienne Bézout, who proved it for polynomials, the identity states the following: Given two nonzero integers $a$ and $b$, let $d$ be their greatest common divisor (GCD). Then there exist integers $x$ and $y$ such that: $$ax + by = d$$ In other words, Bézout’s identity guarantees that we can express the GCD of $a$ and $b$ as a linear combination of these integers. Furthermore, the integers $x$ and $y$ are not unique; there are infinitely many pairs that satisfy this equation. $$ax+by=d$$ $$ax+ab+by-ab=d$$ $$a(x+b)+b(y-a)=d$$ In general $x+nb$ and $y-na$ can be done for any value of $n$. So $x$ and $y$ are not unique. Key points about Bézout’s identity: The integers $x$ and $y$ are called Bézout coefficients for the pair $(a, b)$. These coefficients can be computed using the extended Euclidean algorithm. Bézout’s identity has applications in various areas of number theory

The Extended Euclidean Algorithm is an essential algorithm in number theory. For given integers $a$ and $b$, the extended Euclidean algorithm not only calculate the greatest common divisor $d$ but also two additional integers $x$ and $y$ that satisfy the following equation. $$ax + by =d= gcd(a, b)$$  The existence of such integers is guaranteed by Bézout’s lemma.  Here’s how it works: Problem Statement: Given integers $a$ and $b$, find integers $x$ and $y$ such that $ax + by = gcd(a, b)$. Algorithm Steps: Start with the GCD of  $a$ and $b$. Reverse the steps in the Euclidean algorithm to find $x$ and $y$. Treat the numbers as variables and work our way backward to express the GCD as a linear combination of  $a$ and $b$ Example:  Given $a = 11$ and $b = 3$. We start with the GCD: $$11=3.3 + 2$$ $$3=1.2+1$$ $$2=2.1+0$$ So GCD=gcd(11,3)=1 Now we can reverse the process. Repeatedly replace terms until we get the final result.We have to consider the coefficient of  $a$ and $b$ as variabl

One of the basic techniques of number theory is the Euclidean algorithm, which is a simple procedure for determining the greatest common divisor of two positive integers.First, we need a simple definition: Two integers are relatively prime if their only common positive integer factor is 1. Greatest Common Divisor Recall that a  nonzero $b$ is defined to be a divisor of $a$  if $a=mb$ for some $m$, where $a , b,$  and $m$ are integers.We will use the notation $gcd(a , b)$ or $(a,b)$ to mean the greatest common divisor of $a$ and $b$. The greatest common divisor of $a$ and $b$  is the largest integer that divides both $a$ and $b$ .We also define $gcd(0, 0) = 0$. More formally, the positive integer $c$ is said to be the greatest common divisor of $a$ and $b$ if 1.$c$ is a divisor of $a$ and $b$ . 2. Any divisor of $a$  and $b$ is a divisor of  $c$. An equivalent definition is the following: $gcd(a, b) = max[k, $such that $k | a$ and $k | b]$ Because we require that the greatest common

  Linear Diophantine equations are equations of the form: $$ax+by=c$$ where $a, b, c, x,$ and $y$ are integers, and $a, b$ are not both zero. These equations are named after the ancient Greek mathematician Diophantus , who studied them extensively. The solutions to these equations are integers $x$ and $y$ that satisfy the equation. Linear Diophantine equations have applications in various fields, including number theory, cryptography, and optimization problems. A Diophantine equation is a polynomial equation with 2 or more integer unknowns. A Linear Diophantine equation (LDE) is an equation with 2 or more integer unknowns and the integer unknowns are each to at most degree of 1. Linear Diophantine equation in two variables takes the form of $𝑎𝑥+𝑏𝑦=𝑐$, where $𝑥,𝑦∈ℤ$ and $a, b, c$ are integer constants. $x$ and $y$ are unknown variables. A Homogeneous Linear Diophantine equation (HLDE) is $𝑎𝑥+𝑏𝑦=0$,$x,y∈Z$. Note that $x=0$ and $𝑦=0$is a solution, called the trivial solutio

Modular exponentiation is a fundamental operation in number theory and cryptography. It involves calculating the remainder when an integer, raised to a positive integer power, is divided by another positive integer, known as the modulus. Mathematically, if we have three integers $a$, $b$, and $m$, where $a$ is the base, $b$ is the exponent, and $m$ is the modulus, then modular exponentiation computes: $$a^b \pmod{m}$$ The straightforward approach to calculate this involves computing $a^b$ first and the finding the modulo $m$, which can become impractical for large numbers. However, by exploiting modular arithmetic properties, we can perform the calculation more efficiently. One common algorithm for modular exponentiation is the binary exponentiation method, also known as exponentiation by squaring. This algorithm reduces the number of multiplications required to compute the result. Here's how the binary exponentiation algorithm works: Start with the result $r$ initialized to 1. I