1. Which topics and theorems do you think are the most important out of those we have studied?
It seems to me, and this probably isn't a surprise at all, but I think the big ideas here are: primes. Both the number of primes, and whether a number is prime or not. So, it seems that if I can figure out if a number is prime or not, and how many primes are less than it (or relatively prime) or to be able to prove there are infinite primes, I should be in okay shape.
2. What kinds of questions do you expect to see on the exam?
Again, I expect to see questions really similar to those questions we have seen in homework. Assuming that the final is slightly weighted more towards the material we've covered since the second midterm, I should expect to use different algorithms to show that a number is either prime or composite. I also expect some type of cryptography, as well as some elliptic curve stuff.
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.
Well, I would really like to see number 1 of the last homework assignment worked out. I do not understand the p-1 algorithm at all, and I'm nervous about that.
I also need to work on remembering the names and steps of each primality test. They are all a jumbled mess in my head.
Tuesday, December 6, 2011
Sunday, December 4, 2011
Section 6.3, Due on December 5, 2011
1. What was the most difficult part of the reading for you?
Well, again, there were many things that were difficult. And, admittedly, I'm pretty tired, so I may have missed something. But, specifically I have a question about one thing. I felt like I understood "B-Power Smooth" from the first part of this section... But where I got confused was trying to tell if numbers were B-Power Smooth later on. I mean, from what I understand of the definition of B-Power smooth, it seems like everything is B-Power Smooth to some B... So why is it that in the later parts of this section they'd say that a number "Isn't B-Power Smooth"? Again, I probably just read over what they're saying there, but that's something that confused me.
2. What was the most interesting part of the reading for you?
I know this is going to sound a little lame (I can't help it... I didn't fully understand the reading... so sometimes I have to make stuff up to be interesting) and though I really don't understand this part, I think it's interesting that they can use these elliptic curves to factor big numbers. Seriously, after class I was feeling pretty good and exited about this elliptic curve business, so I'm excited to see what light you'll shed on that subject in class.
Well, again, there were many things that were difficult. And, admittedly, I'm pretty tired, so I may have missed something. But, specifically I have a question about one thing. I felt like I understood "B-Power Smooth" from the first part of this section... But where I got confused was trying to tell if numbers were B-Power Smooth later on. I mean, from what I understand of the definition of B-Power smooth, it seems like everything is B-Power Smooth to some B... So why is it that in the later parts of this section they'd say that a number "Isn't B-Power Smooth"? Again, I probably just read over what they're saying there, but that's something that confused me.
2. What was the most interesting part of the reading for you?
I know this is going to sound a little lame (I can't help it... I didn't fully understand the reading... so sometimes I have to make stuff up to be interesting) and though I really don't understand this part, I think it's interesting that they can use these elliptic curves to factor big numbers. Seriously, after class I was feeling pretty good and exited about this elliptic curve business, so I'm excited to see what light you'll shed on that subject in class.
Thursday, December 1, 2011
Section 6.1-6.2, Due December 2, 2011
1. What was the most difficult part of the reading for you?
Um... Let's blame my confusion on the fact that this is a new book with a new style that feels like a new language being spoken. Yes. Let's blame it on that. I'm still pretty confused about what Elliptical Curves are... but beyond that, I'm confused about what in the world this "Sage" business is. I mean.... what in the world is that? It kind of reminded me of playing Zelda and "Sage" is that annoying helper that beeps and tells you what to do every step of the way... only in this book, since I don't even know what "Sage" is, it's not very helpful to me. But, ignoring the weird Sage business, I didn't understand the condition of -16(4a^3+27b^2) not equaling zero. Also, I didn't understand how figure 6.2 is an elliptic curve... It just looks like dots to me.
2. What was the most interesting part of the reading for you?
Well, I really didn't understand much of what I read... so there wasn't much. I guess that if I had to pick something that caught my eye it would be what the very first paragraph says: It talks about putting keys on stamps and making them short and how these curves help with encryptions... Wow.... I guess it's going to be cool when I understand it...
Um... Let's blame my confusion on the fact that this is a new book with a new style that feels like a new language being spoken. Yes. Let's blame it on that. I'm still pretty confused about what Elliptical Curves are... but beyond that, I'm confused about what in the world this "Sage" business is. I mean.... what in the world is that? It kind of reminded me of playing Zelda and "Sage" is that annoying helper that beeps and tells you what to do every step of the way... only in this book, since I don't even know what "Sage" is, it's not very helpful to me. But, ignoring the weird Sage business, I didn't understand the condition of -16(4a^3+27b^2) not equaling zero. Also, I didn't understand how figure 6.2 is an elliptic curve... It just looks like dots to me.
2. What was the most interesting part of the reading for you?
Well, I really didn't understand much of what I read... so there wasn't much. I guess that if I had to pick something that caught my eye it would be what the very first paragraph says: It talks about putting keys on stamps and making them short and how these curves help with encryptions... Wow.... I guess it's going to be cool when I understand it...
Tuesday, November 29, 2011
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.
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.
Saturday, November 26, 2011
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.
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.
Monday, November 21, 2011
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.
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.
Sunday, November 20, 2011
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.
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.
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