spring 2008 course description
spring 2008 syllabus
guide to giving a math talk
Number Theory, the study of the integers, is the oldest branch of pure mathematics, and also the largest. There are many questions to ask, about individual numbers and their properties, about operations on numbers, about relations between numbers, about sets of numbers, about patterns in sequences of numbers, and so on. Number Theory is famous for generating easy-to-ask, hard-to-answer questions, and that is one reason for its popularity.
Multiplication is the most interesting operation on integers. Number Theory treats factoring and divisibility, and, of paramount importance, prime numbers. The ancient Greeks knew how many primes there are, and Gauss discovered how they are distributed among the integers as a whole. Today we have a few efficient algorithms to find numbers likely to be prime, and some inefficient algorithms to factor large numbers. No one knows a formula to generate primes, nor how many "twin primes" there are.
"Diophantine equations" are equations that specify only integer solutions. For example, the equation 3x + 4y = 5 has infinitely many integer solutions, one of which is (x , y) = (-1 , 2), but the equation 3x + 6y = 5 has no integer solutions. Thousands of years ago, the Chinese knew that there are infinitely many integer solutions to the equation x2 + y2 = z2. Almost 400 years ago, Fermat claimed that the equation xn + yn = zn has no integer solutions when n > 2. He also claimed to have a proof, but it was not until 1995 that Andrew Wiles discovered his own proof and wrote it down.
The Encyclopedia of Integer Sequences, as of this writing, lists over 135,000 different integer sequences that are mathematically interesting. Some of the more familiar are the sequences of even numbers, odd numbers, squares, primes, triangular numbers, perfect numbers, Fibonacci numbers, and Lucas numbers. A less well known sequence is that of the Niven numbers: those divisible by the sum of their digits.
The study of number theory requires an intuition for number relationships and a facility with logic and proof.
The early Greek mathematicians (before 500 BCE) knew only the (positive) integers and their quotients (rational numbers). Numbers were both an area of academic study and a source of mysticism. The discovery of irrational numbers, those that could not be written as quotients of integers, was a major psychological and intellectual blow. Even after this traumatic event, the Greek mathematicians continued to place the integers in an exalted role. Plato says, "Arithmetic" [the study of numbers] "has a very great and elevating effect, compelling the mind to reason about abstract number."
Until recently, Number Theory continued to hold the same pride of place as the most beautiful, most "pure," and least applicable of the fields of mathematics. As both art and intellectual training, it has been part of education and research for thousands of years. The early 20th century mathematician G. H. Hardy took great pride in the belief that number theory was the height of both beauty and uselessness.
Cryptology, the study of encoding messages that could be read only by the intended recipient, is as old as warfare and political intrigue. The obvious and historical methods relied on a two-way key. The writer of the message used the key to encode it, and the recipient used the same key to decode it. Some keys were harder than others to guess, or "crack," but all keys had the disadvantage that they must somehow be communicated between sender and recipient. On the way, they could be stolen.
In the 1970s, a different system of cryptography was invented. In this system, there are two different keys, one to encode, or encrypt, and the other to decode, or decrypt. The encryption key can be public; only the decryption key need be kept private, and only one party needs to have it. This is the kind of system currently in widest use. It was about this time that the internet began to gain importance, and it has become one of the largest users of encryption. It turns out that Number Theory, specifically the branch involving prime numbers, is the major tool in making these encryptions systems work. Number Theory is not so useless after all.
spring 2008 course description
spring 2008 syllabus
guide to giving a math talk
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