3.3:110-115

I didn't understand when they rewrote the function by defining the piecewise function. The numerator made sense, but the denominator seemed a little out of the blue. I am also still having a hard time understanding how it is that convergence helps us to prove these concepts, it seems so unrelated. Even though this class has turned out to be very difficult for me, I suppose it is kind of a good thing. Never before have a gone to every class, done all my homework, studied lots, taken good notes and still not understand. This class is helping me so that I'll have empathy for my future students and previously I have never been in that position.

3.6, due October 28

This section seems relatively more doable than some of the other sections we have been doing lately. First of all, we have already used some of these things like multiplicative functions and we already knew what that means. One thing that was a little bit frustrating is that we were expected to be recalling the Mobius function , but luckily it's not too complicated. I do think it will be SUPER helpful to hear it explained out loud though. What does really confuse me is the inversion formula. I think what's complicated about it for me is combining so many things from different contexts.

3.3--pg 110, due October25

The first part of the reading was a little bit mind blowing. It was weird to look at it as there being 0 primes as x goes to infinity because it shows how spread about the primes get close to infinity. I think that the Dirichlet series mostly make sense although I'm not sure if I understand the underlying concept behind this proof. I mostly get that we're assuming we have finitely many primes congruent to b mod a and then we build a series around that which converges and then show that series diverges. I think I'm not quite getting the part how we build these series. With the Dirichlet characters, I am still trying to figure out what exactly that is and then what we are doing with it. I'm not sure if their notation is confusing or if I'm just looking for more than what is there to be found.

3.2.5, due October 24

As a quick sidenote, the part of class about modular form was entirely over my head...I honestly didn't grasp any of it.
I felt like some of the things in this section weren't defined very thoroughly which made it hard to follow as the section went on. Also, for theorem 3.2.5 it gave a statement and sort of made sense and then said this crazy fact came out of it without giving any explanation about how or why. And, by this point I am very much convinced that there are an infinite number of primes. I suppose it's cool that we can prove it in so many different ways and using completely different areas of mathematics to do so.

3.2.4, due on October 21

On theorem 3.2.4.1 I don't understand why showing that the product is also a sum of four squares means that we only need to prove the theorem for primes. What went on in between here? It's awesome that it somehow means we only need to prove it for primes, at least I thought it would be, but the proof gets crazy after that!

3.2.2, due on October 19

I generally understood the proofs of the lemmas in this section but I think I'm still missing how everything connects for the theorem. Another thing that was hard for me is recognizing why they were saying that certain integers were equivalent to __ mod _ . The beginning of the proof for theorem 3.2.2.2 shows us that if m=uv when u is a sum of squares and v is a sum of squares, then m is also a sum of squares. This seems so simple and I'm surprised I didn't already know it, but I have never before thought about it. This class is helping me to ask questions about things that I have previously accepted for face value and I'm learning to look for patterns and things that are always true which will help me to develop as a math teacher and also help me to aid my students in developing their problem solving skills.

Make up post (Creativity in Mathematics)

I haven't missed a blog yet, but today was a time that I could go to the math forum. So in case I miss one in the future, this is my make up blog.
One of Dr. Chamberland's points that I really liked was that many mathematicians are driven by the aesthetics of math. I feel like I can somewhat identify with this because I have looked at a page of homework that I spent hours on and thought, "man this looks good". Another point he had was that math is not useful if we don't make connections and apply it. Any area of pure math would be completely useless if there was never a connection made between it and some other area. A couple of my favorite quotes from his lecture are, "proofs legitimize intuition", "when the brain is overstimulated, learning is prevented", and "people learn better after a walk in nature than I walk through urban society". Each of these is fairly self-explanatory, but is so applicable to the way each of us studies mathematics.

3.2-3.2.1, due October 17

In the integral quadratic forms, I'm not sure what they mean by substituting integers other than (0, 0, 0, ..., 0) because I don't know where they're suggesting we substitute them. If it means for the ai's then how could it ever be something besides 0? I like theorem 3.2.1.1 because it gives us a general form we can use to find infinitely many pythagorean triples and it's not even that difficult to come up with. I hope that working with polynomials in this class works out better than it did in Abstract Algebra.

3.1.5, due October 14

This section seems a little more doable, although it is just really hard for me to learn from this book. I think most things will make sense once I get to ask questions and we fill in all the parts that the book leaves out. (by the way, I'm asking more questions in class, I hope this isn't annoying-i know we were getting low on time last class). So it seems like we can use any modulus for the lemmas 3.1.5.1-2, if this is true, that blows my mind. I think this section seems to have the most interesting properties of primes/integers that we have looked at up to this point. I'm having a hard time understanding the notation at the bottom of page 81, if we use this, I'm hoping it is explained as it's introduced.

2nd half of 3.1.4, due October 11

First, doing the polynomial division and looking at the patterns last class was really helpful and I'm surprised that the book didn't explain that part.
In the proof of corollary 3.1.4.1, how did they write fn+1=α^(n+1)-β^(n+1) ( I know I didn't write the denominators, but I just don't understand this step). Is there a lot of algebra that they left out here? Or were they using something besides the Binet Formula? I think that the Lemmas 3.1.4.2-4 are interesting because they tell us properties of the Fibonacci numbers that will help us to actually use them to prove things, and just some basic properties so that we can work with them a little bit better (and it's just cool that these things are actually true). 

3.1.4 through the first half of page 72, due October 10

One the "proof" involving the semicircle, I understood the actually math of each step and where they started from but I don't think I quite got the conclusion that they were drawing from it all. I also still don't exactly understand how the fibbonacci numbers are related to the golden ratio, it said that they're related but I never saw how. I'm interested to how the fibbonacci numbers are useful. I've seen them before and I know what they are, but it is such a simple pattern that I've never really seen how they would be useful.

3.1.3, due October 7

First, I have to admit that I am very lost with the new material that was presented on Wednesday. I think it was just too much new stuff, moving too quickly and I couldn't process it as fast as we were going. Also, I spent hours on the homework, and I worked with Ashley to talk things out and I still haven't been able to do any of the problems (one of them I did, but I feel doubtful that it's right). I talked to at least 4 other students who also feel this way. I'm not sure what if anything you can do, or maybe I just need to spend some time having this reexplained to me one on one, but I am really struggling all of a sudden. I really want to do well in this class, and I know I can, but I'm doing the readings, putting in tons of time on the homework, going to office hours, and I'm just not sure what else to do.

This book seems to skip a lot of the middle steps in proofs. For example, I was understanding the proof of theorem 3.1.3.1 but then it didn't say why we had (a^kl +1)/(a^l+1) or why knowing that a^l + 1 divides a^n+1 tells us anything. The proof that is most confusing to me is probably the one involving the GCD of Mm and Mn. The hardest part may be just trying to remember what the Mersenne numbers actually mean because I kind of have to unpack the definition at each step of the proof in order to semi follow it. Which is difficult, since they're new. I kind of like the concept of perfect numbers because I can understand it and it's interesting that anyone thought to investigate such a concept.

3.1.2, due October 4

I'm not sure if I had such a hard time understanding because I just finished taking the test when I read this section or if it was really just that difficult for me to understand. I get the concept that if 1/p diverges, there must be an infinite number of primes, that's simple, and I even understand lemma 3.1.2.1. However, when they introduce the N is where I start getting confused. The Riemann zeta function is conveniently defined so that we can easily just look at the exponent of p and see whether or not the series diverges. I can see why it is nice to have this function defined in such a way, but I don't exactly get their manipulations of it afterward.