I'm sure you're all familiar with Pythagorean triples like
3
,
4
, and
5
which satisfy
32+42=52
. What if we replace the
2
with a
3
? Are there any interesting whole number solutions to
x3+y3=z3
? On the other hand, we've just shown
25
is a sum of two squares. What other numbers can be written as a sum of two squares? Can ever number be written in this way?
I'm sure you're also familiar with equations such as
ax+by=c
, or
x2+y2=k
. We know what their graphs look like, but what can we say about points on these graphs? Are there any whole numbered pairs,
x
and
y
on these curves? How many? What if we allow
x
and
y
to be fractions?
I'm sure you're ALSO familiar with the idea of a prime number, that is, numbers that are only divisible by
1
and themselves. Whole numbers have the fantastic property that we can factor them uniquely into primes. For example,
21=3⋅7
or
2012=22⋅503
. In this way, if we want to understand whole numbers well, it seems like it would suffice to just try and understand these primes. How many primes are there? How are they distributed? Are there patterns to them, or are they sort of randomly distributed? If you can answer this question, you could win $1,000,000!
Such is the beauty of mathematics, with every question we answer, there are always more we can ask! In this class, we'll do our best to answer all of these questions and more!
The style of the class will be as follows. Each lecture, I'll present a sort of conceptual question about numbers or equations, which will motivate the discussion of important concepts in algebra and number theory. Then, we'll finish off each lecture by looking at an important problem in algebra or number theory, and solving it! (most of the time)
Short, fun, and interesting homework problems will be assigned, but they are
optional
and solutions will be given out the following class.
Prerequisites
High school algebra.