Number Theory
Introduction
What is number theory? Number theory is a branch of pure arithmetic, it deals with the properties of integers, mostly positive. It helps with the discovery of interesting relationships between different sorts of numbers to prove that they are true. Number theory is partly experimental and partly theoretical. In history there are two major steps in the development of number theory. The first step was by Euclid who used his Greatest Common Divisor (GCD) algorithm which simplifies fractions into their simplest form using geometrical observation. The next major step of number theory was over two thousand years later by Carl Friedrich Gauss who is referred to as the prince of mathematics.Carl Friedrich Gauss who is regarded as one of the greatest mathematicians of all time. He even went on to state that number theory as “Mathematics is the queen of sciences and number theory is the queen of mathematics.”
How Number Theory is Used
Number theory isn’t always used in day to day life. The mathematicians that study number theory don’t necessarily have a real world application for their research. Most mathematicians that study number theory are fascinated in how numbers fit together and visualizing patterns within the numbers. Mathematicians that study number theory find it fun, but sometimes number theory is used in the world of modern cryptography. Modern day cryptography uses complicated algorithms to encrypt information, and number theory is a major part of being able to do it. Without number theory your data would be easily accessible and would be a major breach in privacy.
Number theory is the foundation of cryptography; without number theory cryptography wouldn’t be as secure and safe as it is today. The best known application of number theory is public key cryptography such as the RSA algorithm. Number theory was and still is a crucial motivation for the development of powerful encryption algorithms and systems. It is also useful in the study of binary codes and other related concepts. Number theory is also heavily used in random number generators. Many principles and ideas that come from number theory are heavily used in computer science, especially coding.
Prime factorization
Prime factorization is a process of factoring a number in terms of prime numbers i.e. the factors will be prime numbers. Here, all the concepts of prime factors and prime factorization methods have been explained which will help the students understand how to find the prime factors of a number easily.The prime factors of a positive integer are the prime numbers that divide that integer exactly. The process of finding these numbers is called integer factorization, or prime factorization. The fundamental theorem of arithmetic says that every positive integer has a unique prime factorization.
What We Learned
Our group learned quite a bit studying number theory, we learned the basics of what number theory is. And we were provided with countless examples on prime factorization and how that is used in number theory. As a group we also learned how to use the Sage calculator to have an easier time factoring larger numbers. We also learned about divisibility theory. One fun divisibility trick we learned is that positive integers are divisible by 9 if and only if the sum of their digits is divisible by 9.