Section 5.4.2, Due on November 30, 2011

1. What was the most difficult part of the reading for you
Well, the reading was short, so I'll make this short. I kind of got lost reading about the RSA Algorithm. I just got lost in the math of it all... with choosing the primes and the e and the... everything. Mostly I felt confused about that, but that could be because I'm just kind of tired....

2. What was the most interesting part of the reading for you?
The most interesting part of the reading for me was reading about the one-way function/authentication stuff. I mostly thought that was interesting because I was trying to figure out how two functions could be public knowledge, and you'd still be able to encrypt/decrypt something. Then, on top of that, to add the "Authentication" seemed kind of genius to me.

Section 5.4-5.4.1, Due on November 28, 2011

1. What was the hardest part of the reading for you?
I felt like this reading was very straight forward. If I had to pick a "hardest thing" it would probably be the trying to remember the linear algebra stuff with matrices. It was just kind of a trip to remember determinants and inverses and all of that kind of stuff. I know it's lame, and not much to go on, but I really didn't have any problem with this reading.... so that was a good thing, I guess.

2. What was the most interesting part of the reading for you?
Well, I'm sure that in the world today we've moved beyond using the simple encryption codes discussed in this section. However, I am kind of a war-movie junkie, and I was incredibly interested to see the roots of coding and to see how messages we most likely coded during early wars. So I think that was cool to read about.

Section 5.3.2, Due on November 22, 2011

1. What was the most difficult part of the reading for you?
Um... I think this will be easy enough. The proof of Theorem 5.3.3.2 (Lucas-Lehmer Test) was the most difficult part of the reading for me. That thing was SO LONG!!!! (... to me...) I'm sure it makes sense to someone somewhere... but for me it just looks like a confusing mess.

2. What was the most interesting part of the reading for you?
I guess, as we've been in these last sections, the things that are most interesting to me are the things that are so modern. I feel like there is some exponential correlation between the year these things were discovered and how difficult they are to comprehend. But, not only that, but I think that it is interesting to view the field of mathematics these days. For example, take a look at the Mersene primes. It almost seems that to be an accomplished mathematician these days, you need to have some sort of computer programing experience. It almost feels like all "Paper and pencil" discoveries are over (obviously, any sort of new proof must be written by someone, and that can't really be done with a computer) and the implementation of new mathematical discoveries (like new Mersene primes) are done by computer programs. We have only to run the programs and wait for the next big discovery. That, to me, is so interesting.

Section 5.3.1, Due November 21, 2011

1. What was the most difficult part of the reading for you?
Well.... Hm... That's a tricky question. I feel like there was so much information packed into this section that I'd struggle to recall most of it. I got the basics for the base stuff and maybe the Carmichael (?) stuff, but after that, as I was reading, I felt like the section was just rambling off information I wasn't ready to handle. Even during the Carmichael stuff, the whole time I was reading it, I thought "Can you just give me an example of what you are talking about?!?" ...Then the did, and I still didn't feel that much clarification from their example. So, I guess that though I didn't grasp the second test they discussed at all, I most struggled with the Carmichael stuff.

2. What was the most interesting part of the reading for you?
 I guess that when we were discussing Quadratic Reciprocity and Jacobi Symbols forever ago (which, truthfully, I'm still struggling to fully understand) I felt like the Jacobi Symbol stuff was so far away and that we'd never reach that section. And, in turn, that was my personal excuse as to why I didn't fully understand it. However, here we are, now reading about the Jacobi Symbols and I am still a little fuzzy about them. I guess that excuse has died down. So, hopefully, if I'm not too busy playing the new Zelda game today, I'll revisit Quadratic Reciprocity and Jacobi Symbols, and hope that helps me out a little bit.

Section 5.3, Due November 18, 2011

1. What was the most difficult part of the reading for you?
I'm sure that you'd expect that I'd get lost at some point... so I'll just say when. I started getting lost during Theorem 5.3.3 (Mostly, I got confused about part (2). But, I'm assuming it's because I need to review that malarky.
After that, I got completely totally completely lost as they started bringing up the AKS algorithm. I mean.... what? It was so so so oh so confusing for me. I'm guessing (hoping) it's because it's so new and stuff. I mean... wow.

2. What was the most interesting part of the reading for you?
Though the part where I got lost made me want to cry, it also fascinated me that something that seems pretty important in the mathematical world has been proven post-2000. Not only that, but it even made it into this crazy book. So, I think that's cool and I want to know how much money that they made for deriving this algorithm.

Section 5.1-5.2 pg 202, Due November 16, 2011

1. What was the most difficult part of the reading for you?
 Well, it must be time to start a new chapter. This reading wasn't too bad. In fact, and don't tell anybody, it was almost kind of enjoyable. However, I did get a little lost in the Proof of theorem 5.2.1. I  mostly think that's just because I'm struggling with the floor function of that proof. I think I just need it verbalized to me. Then, I don't really remember what a kernel is (I remember learning about it, and I think it has something to do with mapping things to zero), so I didn't quite follow the "Square-free kernel of m" stuff.

2. What was the most interesting part of the reading for you?
Well, like I said, I found this reading kind of enjoyable. I think that's just because I had already learned about this technique (the Sieve stuff) in my "History of Math" class a few years ago. So, sometimes when I've tone this homework and I wasn't near a computer to google prime numbers, I'd use this technique to figure out if a number is prime or not. So... That's what I found interesting about it.

Midterm 2 Study Prep - For Real This Time

(First of all, I'd like to apologize again for doing this last week. Second of all, I'd like to really thank you for helping me yesterday via e-mail with those questions. You really helped a lot. The second I read your hint about number 4, I was so embarrassed because I knew exactly what to do... I think my brain is just fried because, believe it or not, I've been studying/preparing for this test all week. I've typed up a study guide from everything you listed on your study guide (don't worry, I know it's not exhaustive) but I thought that might be a good place to start. So... without much further ado, let's get started.)

1. Which topics and theorems do you think are the most important out of those we have studied?
It seems to me that the theme of this exam, as I said last week, would be prime numbers. Specifically, the infinitude of primes. It seems that everything we have done this section has come back around to proving that there are an infinite number of primes.

2. What kinds of questions do you expect to see on the exam?
You know, I really am not sure. Mostly I'm scared. Hopefully we'll see some very similar to the ones on the homework (at least, the questions on the homework assignments that I completed successfully... So just look up my grade and don't ask anything from any of the assignments where I have a zero.)

3. What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Friday.


Characters. Characters. Characters. I'm really praying that you don't put much emphasis on them. They are the bane of my existence. But, I'm guessing they'll be somewhere on the test. So, I really need to work on those. I'm still not sure that I understand EXACTLY what the Method of Infinite Descent is. I skimmed over the section that you recommended, but I probably need to READ it very closely and slowly. However, of everything listed on the study guide, I have been unable to find (I could probably figure it out, after some more studying) in the book, or in my notes, the proof of the second part of the Orthogonality Relations II. I'm just really nervous because it seems that that is one of the proofs that we need to be able to prove, and I don't want to mess this one up. Stupid characters.

4. Are there topics you are especially interested in studying during the rest of the semester? What are they?
I say this with no desire for offense at all. I doubt you'd take any, but I'll just warn you. But, when I was deciding to take this class, graph theory, or combinatorix (?) for my math ed major, I really really wanted to take combinatorix (?) because I'm so interested in that stuff. But, I was nervous that I wouldn't get in, and I wanted my last semester here (wahoo!!!) to be lighter than the rest. So I settled (I know, I'm so mean) for Number Theory. I know I could take combinatorix next semester, but that would require extra money and time I don't have. So I would love to see some more of that. But, I also know that isn't what this class is for. So, I'll try to get over it.

Section 4.2, pgs 144-the end, Due November 9, 2011

1. What was the most difficult part of the reading for you
One of the (many) things that I still don't understand from this (never ending) section is the stuff behind/around/about the constants "A" and now "B". I mean, I understand that they are being used to approximate some information, and that we can throw them around these inequality statements, but what confuses me is following what they are, where they come from, and why they matter. I mean, it just seems that so far they've only been able to give really poor statements about things that we already knew... So... why?

2. What was the most interesting part of the reading for you?
I guess that Bertrand's Theorem was probably the most interesting statement to me. I didn't follow the proof as well as I'd like (it deals with ideas I'm still not totally comfortable with) but just as a statement of n<p<2n, that seems pretty strong and very interesting. I hope to learn more about that, because it really does seem pretty cool.

Section 4.2 pgs 138-144, Due November 7, 2011

1. What was the most difficult part of the reading for you?
Well, first, I must say that while I didn't follow everything in today's reading, it felt worlds better than reading the end of chapter 3. So... hooray. In terms of difficulty, I'll mention a few short things. Do you know that feeling you get when you're listening to somebody (usually a fast talking girl) tell a really long story that has a lot of characters that you're unfamiliar with? At first you really try to keep track of where everybody is in the story so that you can respond appropriately to the punch line? Well, that head-spinning feeling (at first you are okay, but somewhere in the middle you completely lost track but don't want to say anything because you don't want to hurt her feelings) is kind of what I felt tonight. I felt like the proof of Chebychev's estimate was like trying to listen to that girl tell a story. It was a little too long and introduce a few too many new ideas for me to stay caught up. But, as I said, it doesn't feel hopeless. I'm guessing that just a simple explanation/verbalization of some of those "greatest integer" things will be very helpful. Also, I struggled with that "O" business.


2. What was the most interesting part of the reading for you?
(By the way, I'm sorry again for my test blunder... It was a long weekend and I wasn't thinking clearly. But, I did spend most of my Saturday and Sunday preparing for this coming test, so hopefully things'll start looking up) What I found most interesting in this reading was something that you mentioned in class. Mostly, the probability of these large numbers being prime. It reminded me of that Birthday Paradox (I'm sure you've heard of) and how, though (using the book's example) .005% is not a very high percentage, it seems high in context of looking at an 86 digit number and asking if it's prime or not. I mean, 86 digits is a HUGE number, so to have that percent chance of picking a prime seems pretty remarkable. I would have thought that primes were much more sparse than that.

Midterm 2 Study Prep

1. Which topics and theorems do you think are the most important out of those we have studied?
Well, looking back at my notes, it seems like the biggest theme here is: the Infinitude of Primes. It seems to me that there has been one proof after the other how many primes there are. So, I think that any theorems relating to that are going to be the most important.

2. What kinds of questions do you expect to see on the exam?
Like I said in the first part, I'm guessing that we will be expected to proof that there are an infinite number of primes of various forms. Hopefully, we'll get lots of questions about continuous fractions -- I like those. :) Also, perhaps we'll need to be able to calculate certain Mersenne numbers, or Fermat numbers or whatever.

3.What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out in class on Friday.
Well, I've got to say, with very few exceptions, I started getting desperately lost around quadratic reciprocity. I'm thinking that part of that is that I'm missing some real basic understandings somewhere early on, so I feel like I need to return to some of the basics before really delving into the understanding of these sections. But that's what this weekend is for. In terms of a question I'd like to see answered, I guess if you're going to be including anything about characters, I'd like to see the type of question that would be, and how you'd go about answering it. That is, without a doubt, my weak spot for this section. I feel like we just did one theorem after another with out any problems really being solved (I admit, the homework was so whelming to me that I didn't finish any of those assignments). I'd like to see a question relating to those that is solvable (not necessarily a proof).

4.Are there topics you are especially interested in studying during the rest of the semester? What are they?
As I've mentioned before, I'm really just trying to survive and not drown in number theory. I'm interested in any topic that won't make me feel like a complete idiot. Help me Bro. Jenkins. You're my only hope.

Section 3.3 pg 115-end, Due November 2, 2011

1. What was the most difficult part of the reading for you?
OMG, please, Bro. Jenkins, for the love, can we leave this character malarkey behind us forever? I feel so lost and sad and lost. I'm sure that this section (all of 3.3) is very interesting for people who know whatever it is talking about, but I am not one of those people. I do not understand what is happening. Well, I suppose that (after two weeks) I'm almost kind of comfortable with what a character is, but that's pretty much the limit of that. I do not understand L-series, or any of these theorems. It is making me feel so stupid.

2. What was the most interesting part of the reading for you?
 I think it's interesting that we're still reading about these crazy characters. I mean, really. Looking at these proofs and formulas just seem so contrived and crazy. I'm mostly terrified that this concept is going to ruin me, and this class, and my future career, and my life. So, it's interesting that something this silly can be so de-motivating.