Scheme CST 292 Number Theory

 

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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 Func...
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Introduction

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 ...
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Quadratic Residues-Legendre and Jacobi Symbols

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Quadratic Congruence   There is a general question whether an integer $a$ has a square root $\pmod{n}$ and if so, how many are there and how one can find them. We know that the solution of the quadratic equation can be found by  $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ If we want to apply this to the case where $a,b,c \in Z_n$, then $2a$ to be unit in $\pmod{n}$, so that we can divide by $2a$. We can write the equation in the form $$(2ax+b)^2=b^2-4ac$$ If we can find all the square roots $s$ of $b^2-4ac$ in $\mathbb{Z}_n$, we can then find all the solutions $x \in \mathbb{Z}_n$ of the quadratic equation of the form $2ax+b=s$, or equivalently $x=(s-b)/2a$. If we look at the square roots at $\mathbb{Z}_{15}$, then  1 and 4 have four square roots each($\pm 1 ,\pm 4$ and $\pm 2,\pm 7$ respectively, while the other units have none. Example: Find all the solutions of $x^2-3x+2 \equiv 0 \pmod{15}$ given $a=1, b=−3$ and $c=2,$ we have: $b^2−4ac=(−3)^2−4(1)(2)=9−8=1$ So, $b^2−4ac=...
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