By: Mia Bordwin ** **

Number Theory is the study of integers (counting/whole numbers, etc.). Today, it is its own branch of mathematics. While it is one big classification, it can be broken down into many subunits. The following are just some of the sub-classifications; It can be broken down into the Odd numbers sub-classification, which are “those that are not divisible by the number 2”, the Even Numbers-whole numbers positive and negative that can be divided by the number 2- Square Numbers- “numbers that are multiplied by themselves are called square numbers or perfect square numbers”- Cube numbers-“Numbers that are multiplied by themselves 3 times”- Prime numbers- “are numbers that have only 2 factors, 1 and the number itself”- Composite numbers “Unlike prime numbers that have only 2 factors, composite numbers are those that have more than 2 factors”, etc..

**What it is and How it Came to Exist**:

But how did it come to exist? The Number Theory was originally started by the ancient Babylonians. The Sumerians date back thousands of years in their existence. They are known for their advanced civilizations ranging from irrigation systems to a complex postal system. From there, they were able to create the Sexagesimal system, which is a system with a base of 60 which was used for developing writing and counting. They were eventually taken over by the Babylonians after they invaded and turned the capital city into Babylon. With the use of the Sexagesimal system and other advancements, the Sumerians were able to develop what would become a branch of cuneiform, or wedge-based symbols. They used styluses on clay tablets which would then dry in the sun. The Babylonians later adapted this same method. They began incorporating numbers into different aspects of their lives. For example, “The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds.” Two tablets were found on the Euphrates that demonstrated their use of tables to assist in calculations. One of these had the Pythagorean triplets listed. These tables would go on to help solve multiplication, division, and creating equations for algebra. Similarly, the Greeks also noticed a lot about numbers, specifically integers. For example, It is always even when you multiply an odd number by an even one. Later mathematicians would build off of the preexisting math discoveries, like Carl Friedrich Gauss, who “demonstrated an analysis of the factorization of polynomial equations…and contributed to the understanding of the prime number theorem.” It wasn’t long before mathematicians would be used in astronomy, music, etc., that would all come to shape its existence.

**Use of Number Theory in Cryptography**:

The Number Theory influences so many aspects of our lives and branches of study. However, one of the biggest influences it has had is on cryptography. As explained by Kaspersky, “Cryptography is the study of secure communications techniques that allow only the sender and intended recipient of a message to view its contents. The term is derived from the Greek word *kryptos, *which means hidden.” In other words, cryptography is the scrambling of letters and numbers to send hidden messages between users. As the world is becoming more technologically advanced and dependent, the use of cryptography is becoming more and more influential. For these reasons, the RSA cryptosystem was created. The RSA algorithm is a branch of public key cryptography. It is named for the three MIT researchers, Ronald Rivest, Adi Shamir, and Leonard Adleman, who published the cryptosystem. The main component is number theory. Especially, “applications of prime numbers, modular exponentiation, and Euler’s Theorem [which are] are integral to successfully implement the RSA cryptosystem.” In simple words, the system works by the users converting their messages into numbers. The numbers are not random, but rather follow a table known as the ASCII table, or American Standard Code for Information Interchange. The number theory permitted the creation of the table, along with the system.

To further, cryptography is becoming increasingly useful in the cryptocurrency world. For background context: Cryptocurrency is a digital currency that was originally created using encryption algorithms. In order for cryptocurrency to function effectively, it was created on cryptography and blockchain technology. Blockchain technology is defined as “a system of recording information in a way that makes it difficult or impossible to change, hack, or cheat the system.”

Therefore, it is necessary for the messages and transactions sent to be built based on, for example, the ASCII Table. This has helped make cryptocurrency prominent in our society today. Whether it be the fact that messages and transactions are transparent or even that it is free from censorship, it now offers many benefits to its users. However, because of its anonymity/transparency and lack of traceability, it has become a preferred network for illegal activity. Number Theory has helped shape its existence, creation, and profound impact on the world’s economics and security.