1 Introduction
Numbers have both fascinated and been utilized by civilization for thousands of years, beginning with their ability to quantify a number of objects. Over time, mathematics evolved into several branches, one of them being Number Theory. In Britannica, Dunham (2022) defines Number Theory as "the branch of mathematics concerned with properties of the positive integers (1, 2, 3, . . . ). Sometimes called `higher arithmetic,' it is among the oldest and most natural of mathematical pursuits."
By DeFacto
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Before the invention of computers in the 20th century, Number Theory had very little application to the real world. Ironically, the inability to be applied is often labeled as "pure" in mathematics, so Number Theory was thought to be the "purest" of all mathematics. World World War II changed much of this view point. Mathematics was an essential tool for breaking the German's Enigma Code and many other ciphers ("Cracking the Code", 2019). At Bletchley Park, Brittan's secret headquarters for code-cracking, the first large-scale electronic computer named Colossus was created to crack the lesser-known German "Tunny" code according to Britannica (Copeland, 2017). The code-breaking of WWII lead to the advent of digital computers and digital communications. The theorems of Number Theory began to provide unexpected answers to real-world problems, which first began with the field of cryptography.
Cryptography, as defined by Hosch (2022) in Britticiana, is the "practice of enciphering and deciphering of messages in secret code in order to render them unintelligible to all but the intended receiver." Due to its existence for over 3,000 years ("Cracking the Code", 2019), there have been countless forms of cryptography. The History and Background section focuses on two early forms of cryptography—the Caesar Cipher and Vigenere Cipher—which sets the stage for some fundamentals of modern Public Key Cryptography. There are many ciphers not mentioned, such as the Diffe-Hellman Algorithm.