# history of analytic number theory

Given two integers d 6= 0 and n, we say that d divides n or n is We glossed over the specific steps involved in calculating the GCD for our example, but, hopefully, the illustration above provides an intuitive understanding of the geometry involved. His masterpiece publication, Disquistiones Arithmeticate (loosely translated to “Arithmetical Investigations) packed multiple brilliant & precise methods that, while not necessarily all his original work, aggregated & systematized the field of Number Theory. The first, states that any integer greater than 1 is either prime itself or can be constructed by multiplying strictly prime numbers. These are now called Fermat primes.

In other words, what’s the great common divisor of 15 & 25?

https://www.cs.purdue.edu/homes/spa/courses/cs182/mod5.pdf, Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. Soon thereafter, he established a beautiful result known as Wilson’s theorem: p is prime if and only if p divides evenly into [(p−1) × (p−2) × ⋯ × 3 × 2 × 1] + 1. Number Theory is at the heart of cryptography — which is itself experiencing a fascinating period of rapid evolution, ranging from the famous RSA algorithm to the wildly-popular blockchain world. By this contradiction, Fermat concluded that no such numbers can exist in the first place. For instance, 39 ≡ 4 mod 7. The same Dirichlet (who reportedly kept a copy of Gauss’s Disquisitiones Arithmeticae by his bedside for evening reading) made a profound contribution by proving that, if a and b have no common factor, then the arithmetic progression a, a + b, a + 2b, a + 3b, … must contain infinitely many primes. By contrast, number theory seemed too “pure,” too divorced from the concerns of physicists, astronomers, and engineers. The first published statement which came close to the prime number theorem was due to Legendre in 1798. 1800 BCE) contains a list of "Pythagorean triples", that is, integers \$\${\displaystyle (a,b,c)}\$\$ such that \$\${\displaystyle a^{2}+b^{2}=c^{2}}\$\$. Euler gave number theory a mathematical legitimacy, and thereafter progress was rapid. In order to minimize costs, we only want to buy tile length of the same size; which requires that we calculate the largest length of tile (in meters) that’ll perfectly fit, both in length & width, without breaking apart. Despite Fermat’s genius, number theory still was relatively neglected. Calculus is the most useful mathematical tool of all, and scholars eagerly applied its ideas to a range of real-world problems. And when he turned his attention to amicable numbers—of which, by this time, only three pairs were known—Euler vastly increased the world’s supply by finding 58 new ones! Now equipped with the basic history of number theory & a quick preview into the depth of its impact, it’s time to familiarize ourselves with the most applicable topic within number theory: cryptography. The prime number theorem then states that x / ln(x) is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1: As we’ll see next, while Gauss formally set the stage for the branch, early examples of cryptographic systems were already well in existence, with pretty daring stakes. For example, π(10) = 4 because there are four prime numbers (2, 3, 5 and 7) less than or equal to 10. The Wikipedia definition above becomes digestible by splitting it into two separate parts. Often inviting our greatest thinkers to unravel the many, deep mysteries of the cosmos, the study of natural numbers, Number Theory, is one of the oldest branches of mathematics. An extraordinary mathematician, Euclid of Alexandria, also known as the “Father of Geometry,” put forth one of the oldest “algorithms” (here meaning a set of step-by-step operations) recorded. 0.1 Divisibility and primes In order to de ne the concept of a prime, we rst need to de ne the notion of divisibility. He used a technique called infinite descent that was ideal for demonstrating impossibility. For three and a half centuries, it defeated all who attacked it, earning a reputation as the most famous unsolved problem in mathematics. The cornerstone eureka moment of Disquistiones is a now-timeless theorem known as the Fundamental Theorem of Arithmetic: Any integer greater than 1 is either a prime, or can be written as a unique product of prime numbers (ignoring the order). Take a look, How to do visualization using python from scratch, 5 YouTubers Data Scientists And ML Engineers Should Subscribe To, 21 amazing Youtube channels for you to learn AI, Machine Learning, and Data Science for free, 5 Types of Machine Learning Algorithms You Need to Know, Why 90 percent of all machine learning models never make it into production. Of course, even Euler could not solve every problem. Here are a few examples: Uncharacteristically, Fermat provided a proof of this last result. This algorithm, the Greatest Common Divisor, stands the test of time as our kickoff point for Number Theory due to the fascinating properties it highlighted in natural numbers. The following large leap in Number Theory stems from a break-through approximately ~2000 years after Euclid. Initially, Euler shared the widespread indifference of his colleagues, but he was in correspondence with Christian Goldbach (1690–1764), a number theory enthusiast acquainted with Fermat’s work.