Math 317: Elementary Number Theory

Math 317: Elementary Number Theory

Why is it fun/useful?

Number Theory is a branch of mathematics, which aims to discover many deep and subtle relations between different kinds of numbers. Positive integers, for instance, from the beginning of recorded history has inspired a great fascination- aesthetic, mystical and practical!

The following is a story told by G. H. Hardy, an eminent British number theorist (1877-1947).  He collaborated with S. Ramanujan (1887-1920), a brilliant Indian mathematician. Once Hardy visited Ramanujan who was ill in a hospital in England and on his arrival he remarked that the license plate of the taxi he had taken was 1729, which seemed uninteresting. Ramanujan immediately replied that, on the contrary, 1729 was very interesting, being the smallest positive integer expressible as a sum of two positive cubes in two different ways, namely 1729=10+9=12+1.

Having heard this story one might be interested to find out if there are only finitely many integers expressible as a sum of two positive cubes in two different ways! Or one may ask if one can characterize the integers expressible as a sum of two positive cubes in at least one way. Or, are there any cubes, which are sums of two cubes?

This course exhibits many amazing properties of integers and deals with a whole range of problems, from Pythagorean Theorem to applications in Cryptography. It uses only very basic mathematical knowledge, and on the other hand introduces the student to abstraction and proof techniques in a very gentle manner! Hence any student with an interest in mathematics would enjoy it, while students planning to take more abstract courses like Math 301 or Math 311 will find it very beneficial.