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MAT 338 Course Notes Day 1
A first problem
Let's start! Suppose you have as many postage stamps as you want numbered 2 and 3. What
denominations of postage can you get by combining those? (Or, perhaps, what den
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MAT 338 Course Notes Day 8
Prelude
Here are three interesting problems which may seem totally unrelated at first:
You have lots of volunteers at a huge campaign rally. Becau
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MAT 338 Course Notes Day 12
Prologue: Why congruences matter
Did you find a pattern?
Before we get to groups, let's take a final look at whether you can find a pattern in the primes p
such that x^2\equiv -1\textcfw_ mod (p) has a sol
MAT 338 Course Notes Day 15
More fun with Phi
There were two somewhat theoretical things on the previous exercises. They are worth talking
about briefly.
A formula for Phi
First, a formula for
(N)=(k=1npeii)
Did you get one? Here it is!
(k=1npeii)=k=1n(pe
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MAT 338 Course Notes Day 14
More exploring of Phi
Trying things out
One of the neatest things about (n) is that it is quite useful for things we are familiar with
(congruences) but is also a prototype for the many functions there are
MAT 338 Course Notes Day 10
Redux of homework
We have seen some neat stuff regarding Zp now! And there will be a lot more to come. We now
wrap up some preliminary ideas, so that we can forge ahead into specific questions about Zp
soon.
One homework questi
MAT 338 Course Notes Day 9
The integers modulo a prime
Let's introduce ourselves to Zp! After all, this friendly number system will become a good
acquaintance, if not friend, throughout the rest of the course. We'll explore it soon, but first let's
see so
MAT 338 Course Notes Day 11
Essential Facts about Groups for Number Theory
Many of the bookkeeping issues which arise in number theory can be made much easier by
changing our language and introducing a small amount of abstraction. That abstraction is the
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MAT 338 Course Notes Day 4
Followup
First, recalling the homework, I just have to be gratuitous; try evaluating this next cell, and
seeing whether you can get a GCD which is not 3.
1
@interact
2
def
MAT 338 Course Notes Day 5
An interesting coincidence
We spent the last couple times discussing linear and quadratic Diophantine equations. As you can
see, even relatively simple questions become much harder once you have to restrict yourself to
integer s
MAT 338 Course Notes Day 3
Complements to Bezout
There are several interesting things that the linear combination version of the gcd can give us.
Relatively Prime
Remember, a and b have greatest common divisor 1 precisely when you can write ax+by=1 for
so
MAT 338 Course Notes Day 7
Review
A homework proof
Here is an example of one of the homework proofs, just to give you the idea.
Let's prove that if a3b2, then ab.
By the Fundamental Theorem of Arithmetic, we know that a and b may both be written
as a prod
MAT 338 Course Notes Day 6
Redux of the FTA
More thoughts on redoing theorems
The first problem on the homework was a little interesting. Did you all get the obvious solution?
1
end_range=100
2
[(n,m) for n in range(1,end_range) for m in range(n+1,end_ran
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MAT 338 Course Notes Day 2
Welcome back!
First, let's test some conductor ideas. In the cell below, which should automatically evaluate for
you, Sage will automatically list all the nonne
MAT 338 Course Notes Day 16
Exercise Review
Divisibility and Phi
Our first item for today is just the coolest result. I hope you conjectured it, but it could be quite
challenging to do so. Yet once you see it, it seems obvious to at least try it!
Proposit