Number Theory

What are numbers made of? Beginning with a study of the integers and congruence, this question will lift us into the realms of quadratic reciprocity, RSA encryption, and beyond!

The whole numbers (integers) are something that most of us are pretty familiar with. When it comes to deciphering their underlying structure, however, this seemingly straightforward task has presented a challenge for some of history’s greatest mathematical minds, from Pythagoras to Fermat to Euler to Andrew Wiles. The questions they have sought to answer are often remarkably simple to state but their solutions, on occasion, can lead to deep results in geometry, analysis and various other areas of mathematics. We will aim to lay down the theory needed to begin thinking about these questions rigorously and we will start developing some of the tools that modern number theorists have used and continue to use to tackle problems of this sort. This will initially involve studying the axioms of the integers, unique factorization into primes and the Euclidean Algorithm for calculating the greatest common divisor of two integers. Then we will quickly progress to congruences, Fermat’s Little Theorem (not his Last) and the number theory behind RSA encryption which is central to internet security!

One of the central goals of this course is also to introduce you to mathematical proof and the way in which professional mathematics is done. No familiarity with proof is required to begin, however, only an enthusiasm for mathematics and for learning how to solve (and ask) fascinating questions!

For the application...

Prerequisites

1-2 years of high school algebra. Some exposure to calculus is desirable, though not a rigid requirement.

Relevant experience

Tell me about the mathematics and science classes you have taken so far as well as any competitions you may have taken part in. Also, if you have ever been interested enough to just further investigate some idea from one of your classes or to read more about it outside class, please write about that too!

Application Question (Core-specific free response)

(1) Substitute $$x = y - 2$$ in the following cubic expression and simplify to get another cubic expression with no quadratic ($$y^2$$) term:

$$ 2x^3 + 12x^2 - 3x + 1 $$

(2) Confirm that

$$ (n^3 - 3n)^2 + (3n^2 - 1)^2 - (n^2 + 1)^3 = 0 $$

(3) Find some examples of sets of three integers, $$x$$, $$y$$, and $$z$$, such that $$x^2 + y^2 = z^3$$.