**Author: Afreen Hossain**

**Introduction:**

Number theory is a fascinating branch of mathematics that deals with the properties and relationships of integers. It may sound intimidating, but at its core, number theory explores the fundamental nature of numbers. In this beginner's guide, we'll take a journey through the basics of number theory, unraveling the mysteries that lie within the world of integers.

A few topics under number theory are:

**The Foundation: **

*Integers*

Let's start with the basics. Integers are whole numbers, both positive and negative, including zero. They form the foundation of number theory. Examples of integers are -3, -2, -1, 0, 1, 2, 3, and so on. Number theory focuses on understanding the unique properties and patterns within this set of numbers.

*Divisibility and Factors*

A key concept in number theory is divisibility. An integer 'a' is said to be divisible by another integer 'b' if 'a' can be expressed as 'b * c', where 'c' is also an integer. For example, 15 is divisible by 3, as 15 = 3 * 5.

Factors are integers that divide a given number without leaving a remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. Number theory delves into understanding the properties of these divisors and how they relate to the integers.

*Prime Numbers*

Prime numbers are a crucial element in number theory. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Examples include 2, 3, 5, 7, and 11. Every positive integer can be uniquely expressed as a product of prime numbers, a concept known as the Fundamental Theorem of Arithmetic.

*Greatest Common Divisor (GCD) and Least Common Multiple (LCM)*

The GCD of two integers is the largest positive integer that divides both numbers. For example, the GCD of 8 and 12 is 4.

The LCM of two integers is the smallest positive integer that is a multiple of both numbers. For instance, the LCM of 8 and 12 is 24. These concepts are fundamental in solving problems related to divisibility and factors.

*Modular Arithmetic*

Modular arithmetic is a fascinating aspect of number theory that deals with remainders. In modular arithmetic, we work with the remainder when dividing one number by another, known as the modulus. It has applications in cryptography, computer science, and various other fields.

*Quadratic Residues*

*Quadratic Residues and Non-Residues*: In modular arithmetic, quadratic residues are squares of integers that leave the same remainder when divided by a particular modulus. Non-residues are numbers that are not quadratic residues.

*Law of Quadratic Reciprocity*: A fundamental result in number theory that establishes a relationship between the solvability of quadratic congruences with different moduli.

*Arithmetic Functions*

*Euler's Totient Function*: Counts the positive integers up to a given number that are coprime (have no common factors) with that number.

*Möbius Function*: A function defined on the positive integers with applications in number theory.

*Ramanujan's Sum*: A type of series discovered by the Indian mathematician Srinivasa Ramanujan.

*Additive Number Theory*

*Partition Theory*: Deals with ways of expressing a number as the sum of positive integers.

*Goldbach's Conjecture*: Posits that every even integer greater than 2 can be expressed as the sum of two prime numbers.

*Waring's Problem*: Explores the representation of numbers as sums of powers of positive integers.

*Elliptic Curves*

*Elliptic Curve Arithmetic*: Studies the properties of elliptic curves and their points.

*Applications in Cryptography*: Elliptic curve cryptography utilizes the difficulty of solving certain problems related to elliptic curves for secure communication.

*Cryptography*

*RSA Algorithm*: A widely used public-key encryption method based on the difficulty of factoring large composite numbers.

*Diffie-Hellman Key Exchange*: Allows two parties to establish a shared secret key over an untrusted communication channel.

*Applications of Number Theory in Modern Cryptography*: Utilizes number theory concepts to ensure the security of cryptographic systems.

These are only a few illustrations of the wide topic of number theory, which has numerous links and is still being studied. It can be used in many branches of information theory, computer science, and cryptography in addition to pure mathematics.

**References:**

YouTube: Home, 9 November 2017, https://www.programmersought.com/article/22024550501/.

Number Theory and Cryptography I. Introduction, https://pi.math.cornell.edu/~mec/2008-2009/Anema/numbertheory/intro.html.

YouTube: Home, 9 November 2017, https://www.profaccred.com/number-theory/.

YouTube: Home, 9 November 2017, https://www.vaia.com/en-us/explanations/math/pure-maths/number-theory/.

“Number Theory - Definition, Examples, Applications.” Cuemath, https://www.cuemath.com/numbers/number-theory/.

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