section 5.4.2, due November 30

The RSA algorithm was difficult for me to follow. I think I understood the very basic concept, but I got lost on most of the details. It's cool that all the things we've learned about primes come into play with this algorithm. I'm wondering how pseudoprimes and other special cases affect this code. Are they wise to choose? Unwise? I'd probably be able to figure that out myself had I understood it a bit better.

5.4 through 5.4.1, due November 24

This section was really interesting and I almost just kept reading. The only part that I didn't quite understand was example 5.4.3, it didn't seem like it was really explained. It's cool to see an area where number theory is applied because there is so much of the class that is purely abstract in my mind and I sometimes can't think of a context where it might be used.

5.3.2, due November 22

First question: does the Lucas-Lehmer test always work for Mersenne primes? (It sounds like it does.) This proof is a great idea but it is far from intuitive and the proof was extremely difficult for me to follow. I know we already talked about this, but that is a huge number that we know is prime! It's cool that mathematics has so much left undiscovered. I think that's one reason I love it so much.

5.3.1, due November 21

In the example that went along with the first definition, is the book supposed to say that 25 is not a pseudoprime base 2 or 3? Because according to the definition you work mod n when testing n's primality. So it doesn't make sense to talk about using a different number there. So how many bases do we have to test to decide that n is pretty likely to be prime? Is there a general rule for this? And how would that work since we have lemma 5.3.1.1 where every time we test one more base, we're getting many more. Since we know that a Carmichael number must be divisible by three primes, it will never be prime. But, depending on how many units there are mod that number, it could have satisfied fermat's theorem for a lot of bases. Does that mean that purely looking at the number of bases that work for a number gives no good indication of primality?

5.3 up to section 5.3.1, due November 17

I had no idea the numbers that are considered to be "small" for a computer were actually so big! When they were using Fermat's theorem to see if a number could be prime, I got confused because they were looking to see if 2^11386 was 1 mod 11387, but it didn't look like they ever actually found that because they had 2^11387. It's easy to find from there but why did they stop there? At first I was all excited because someone made fermat's deterministic, but as soon as I read it I was disappointed cause it'll still take forever. This concept is a little frustration because we have proved a million ways that there are infinitely many primes and yet we don't know how to find very many of them.

5.1 and section 5.2 through the top of page 202, due November 15

I'm glad we're finally getting closer to being able to effectively test whether or not a number is prime. This section is interesting because it gives insight into how computers test for prime numbers.  The sieve method just looks like what anyone naturally does when checking for primes. I guess it's nice because you end up with a compiled list rather than doing the process for each number. The next example was a little bit confusing to me because I thought they were finding primes again but then it made more sense when I realized that they weren't trying to do that. This was about as far as my understanding extended. Theorem 5.2.1 didn't make sense to me and I think it is partly because there is the greatest integer function and a sum (i'm tired of doing sums). I can compute numbers in this formula but I don't follow the proof because I don't understand how to interpret it.

Test 2 questions, due November 11

I think the theorems that define functions are important. Also, the ones the required about 20 lemmas in order to prove. I have no idea what to expect from the exam other than a few problems that are basically computation and then 1 or 2 proofs. I need to work on understanding everything. What we've been doing recently and the proofs where we switched summations are still confusing to me. Any of the review questions will help me. For the rest of the semester I just really hope we do something that is at least accessible. It is so frustrating to be in class every day, do the readings each night, come to office hours every day, and still the material is just flying over my head. I want to understand so badly and I just don't know how.

the rest of section 4.2, due November 8

On theorem 4.2.4, I need some help with the notation and understand what it's saying. I think that Bertrand's theorem is telling us something really awesome about primes and how their density is related to where in the natural numbers we are looking (really big numbers or pretty small numbers). I think it'll be cool to talk more about the consequences/interpretations of this theorem in class. Or maybe I'm just reading too much into it.

4.2 through the middle of page 144, due November 7

I have never seen an approximation for factorial and I've never thought about one being useful because I often forget to think about these concepts to really large numbers where it would be difficult to compute them. I'm curious how good of an approximation it is. I think the following lemmas might be getting at that, but I'm a little confused about where they came from (even with the proof) and how to interpret them. I have really lost sight of the big picture, I know we talked about Chebychev's estimation last time and so of course we read the (very confusing) proof today, but the other things seem somewhat random. I know they somehow connect because we keep learning lots of lemmas in order to prove theorems, but they still seem out of place to me.

4.1 and section 4.2 through page 138, due November 4

In the first section, I could easily see that these different functions had similarities and a connection should be drawn but I didn't follow their explanation very well. I'm glad we're finally looking at the binomial theorem cause I'm hoping that this time it will finally stick! and I am excited to better understand pascal's triangle because I was introduced to it in 9th grade and didn't see it enough to gain much from it besides that.

The rest of 3.3, due November 1

First of all, I don't think I have any idea what the orthogonality relations are. I know this doesn't exactly help you know what to teach tomorrow, but I think my biggest problem right now is that we are building each day but there are always things I'm still not getting. The other thing is that the logic used in these recent proofs seems very different  to me than what we have done in the past. When 3 pages are spent on a proof and I get lost near the beginning, it is incredibly difficult for me to follow the rest of it, especially because I haven't been able to see where these are going. My favorite part of this section was that they didn't prove the last theorem. Yay!