Section 6.1, due December 2

I found it a little confusing why single points were bad in our scenario. It seems fine to add a single point to itself. It may not be a very interesting curve, but I could not see why it would be banned. Also, I did not quite understand what characteristic 2 meant.

However, I did find it interesting that elliptic curves are all abelian groups. We know abelian groups are really just products of additive groups in Zn, so it seems that we could just classify elliptic curves as such. It may just be too hard to map the points on the curve to their corresponding points in the products of Zn. However, if this was done, it would seem to open up all sorts of opportunities.

Section 5.4.2 due November 30

In reading about the public key systems, it was a little confusing at first all the notation associated with RSA. After they gave an example, it became much clearer. I think the part that threw me at first was the different bases for the numbering system. I thought it was going to do it all in base 26 or so, but it makes sense that it is easiest to convert to base 10 first.

I did find it fascinating that this area seems to be changing. It is like those trying to encrypt are always on the lookout for better algorithms, and they hope to find them before that other party can figure out a method to decrypt too fast. Another part that was interesting was how this depends on a problem taking a very long time to solve relative to the creation of the problem. After all, to encrypt something, it needs to be quick, but those trying to break the code have much more time to tear it apart.

Section 5.4 due November 28

I found it difficult at first to understand the matrix cipher, but after seeing the example it made quite a bit more sense.

I found all the various methods for encryption to be interesting. There are just so many ways, but some of them seem to be rather vulnerable. I wondered why they would include the simple examples like the Caesar cipher, since it is easily broken, but I guess like most things, it is important to start with the basics before getting down to the tough stuff.

Section 5.3.2 due November 22

The proof for the Ludqw Lehmer test was a little difficult for me to understand. I did not follow the proof very well. It was bouncing around in all sorts of areas, and in general just hard to see where it was going. However, it was rather interesting how it brought in so many different concepts.

Another part that was interesting was how it revolves around the Mersenne primes. They seem to be a particularly useful breed of primes. Also, it seems that they would be 3 mod 4 which we discussed in cryptography to be a desirable trait in a prime number.

Section 5.1, 5.2 due November 15

I found it a little confusing how Legendre's Formula worked. I could not quite follow what it was doing or how exactly it worked. From what I did understand, it was there to find numbers relatively prime to m. I figure if one multiplied a bunch of primes together, and then used this method, they could find a fair number of primes.

Something that I found interesting was the general idea of testing for primes. The methods that were introduced seemed mostly like those that I learned in grade school. However, I imagine that it will get more sophisticated. Another side note, showing that the sum of twin primes converges was rather interesting as well.

Exam Review, due November 11

I feel that the major portion of this section was spent handling how there are an infinite number of primes, and also Dirishlet's theorem (some of the simple cases). Also, quite a bit of time was spent discussing arithmetic and multiplicative functions as well.

I would expect some proofs for some general concepts that we have covered (similar to the homework problems). I would also expect some computational problems such as continued fractions or Fibonacci/golden ratio questions.

I would like to go over a little bit of the concept behind sums of squares problems. Just section 3.2 in general.

I would like to go more into how to find primes in general.

Section 4.2, due November 9

I found it a little tough to follow Bertrand's theorem. The logic was a little difficult to follow and all of the simplifications sort of got jumbled in my mind. I could not quite follow where it was going or how it was arriving at certain conclusions.

However, I did find it interesting how we can set so many bounds on the number of primes, how big the primes are, and some sums of primes as well. It seems that we can say quite a bit about the distribution of primes whereas I figured it was more or less random.

Section 4.2, due November 7

Reading through the proof of Chebechev was a little confusing. It was difficult to follow all of the algebra and substitutions that were made. It makes more sense looking over it again, but still, it was tough with all the identities used and substitutions.

I have heard of big and little o before, and I remember it having something to do with the time it takes to solve a particular problem. It was also connected with the unsolved p = np, but I am not completely sure. Perhaps I am piecing two things together which are actually unrelated.

Section 4.1-4.2, due November 4

For the most part, this section made sense. The one part that was a little confusing was that the inverse of x/ln(x) is an approximation to p(n). However, looking on it a little more helped make more sense from it.

Something interesting was the need for complex analysis at first to prove the prime number theorem. I found that very interesting considering how I would, before this class, not consider how the complex numbers and integers are related. Perhaps it goes back to those modular forms taking numbers in N and mapping them to elements of C.

Section 3.3, due November 1

I did not follow the last part of the proofs. It was a little intense in their calculations, and I could not quite understand how they were matching up to one another.

However, the result of the theorem is rather exciting. To me it sounds like if I draw a line with the y intercept not being 0, I will find my line to intercept an infinite number of primes for y. That is rather interesting. It just goes to show, there are lots of primes out there.

Section 3.3, due October 31

I actually did not understand most of this section. the L-series was rather confusing. Its definition was straight forward, and the proofs were as well for the most part, but I suppose I just do not see where we want to go with this. However, I guess I should mention that the computations were fairly straight forward.

One interesting part was the connection the L-series has with the Reimann Zeta function. They look almost identical (except for the character). I suppose that is where it is heading.

Section 3.6, due October 27

Something a little rough about this section was the Mobius inversion theorem. The symmetry of d,k was a little much for me. I could not completely understand how that worked or why. I need some more processing time on that one.

Something interesting was how multiplicative functions work. It seems to provide quite a bit of power in proving things when one knows that the function is multiplicative. Much more can be said about the function and the things it describes.

Section 3.3, due October 26

I understood for the first bit what was going on, but then it started to get more and more complicated as to what the characters were doing. I do not think that I completely grasp what a character actually is. I sort of went between it being something congruent to k, and a complex number.

However, I did find that the phi function was useful once again. It seems to show up quite a bit. In fact, anything we do modulo something or another seems to be connected to it. I do not see any connection between the most recent sections, but I am sure it will present itself in the upcoming sections.

Section 3.2.5, due October 24

The lemma 3.2.5.1 was a little confusing. I was not completely sure what it was saying. It was talking about the period, which I sort of understood to be how long until the sequence repeated. however the A_i, B_i, I could not figure out what that really was, or how it played into the following theorem.

However, one interesting aspect was how every infinite continued fraction was an irrational number. It makes sense that it could not be rational, but I had never thought about this before. I have seen continued fractions elsewhere, but I am still just starting to get a taste of what they are good for.

Section 3.2.4 Due October 20

I did not completely follow the proof of every integer being written as the sum of 4 primes. I could not really understand why they were doing what they were doing. However, upon reading it over, it started to make a little more sense.

I think it is cool how you can write any integer as a finite sum of powers. That is certainly an interesting result. I do not know what it would lead to, but I never thought such a thing was possible.

Section 3.2.2 due October 19

I did not understand all of the equivalence statements on r(n) the number pairs that satisfy a^2 + b^2 =n. I do not really understand its importance, or how one could calculate it efficiently. I imagine it will make more sense in the next section.

I did find that p=1 mod 4 to be rather interesting. Something about mod 4 makes a huge difference and it seems to be stemming back to quadratic reciprocity. It sort of seems like a super odd or something like that. Most of the time, it seems that if something is odd verse even lots can be said, but this here is breaking odds into two categories and one does quite a bit, the other the opposite.

Section 3.2.1, due October 17

One concept which was a little difficult for me was following the proof for x^4 + y^4 = z^2. It made more sense when I saw the contradiction, but because I was not quite sure where they were going, it made it a little harder to follow. Perhaps I should read the last couple of lines of the proof to get the general outline.

One part I found interesting was how the Pythagorean triples could be represented with 2mn, n^2 - m^2 and n^2 + m^2. It made sense, but I had just never seen it represented in this way before. It would have been nice to know this result years prior...

Section 3.1.5, due October 14

I found the last proof of there being a prime divisor congruent to 1 mod m of m^m - 1 to be a little confusing. It might take a bit to digest that one. Just trying to follow all of the logic was tough for me. I am not quite sure why they were taking products of m/primes and so forth. I might just have to look at it again.

However, the others proofs of proving that various congruences had infinitely many primes in them was also exciting. I found it interesting that with the same modulus, the variety of methods used to prove the different equivelances to be impressive. It was interesting that one size did not fit all.

Section 3.1.4, due October 12

I started to struggle to comprehend exactly what was going on in the end of the section. It started to use the Binet formula, which I really have no idea what that is. Just keeping track of all the things the Fibonacci numbers do was a little tough.

However, the connections the Fibonacci numbers have is quite impressive. They play in well with many things such as the golden ratio, and primes. I was amazed to see the wide variety of places they have a presence in.

Section 3.1.4, due October 10

To me, I found it a little difficult to understand exactly how the golden ratio was being constructed from the ruler and compass. I could not quite follow what they were saying.

However, I did find the applications of the golden ratio to be fascinating. I had heard about the rectangle, but I had no idea the idea extended to so many geometric applications. I think I understand a little better why it is called the golden ratio. It shows up over and over in more regular and images with some sort of symmetry.

section 3.1.3, due October 7

I did not quite understand the Euler part of the Mersenne numbers, but as I read through it more, I think I am starting to get it better. Overall, it was fairly straight forward, but I am not quite sure where we are going with all of these lemmas.

I did think it was interesting how Mersenne primes are related to perfect numbers. I would not have connected the two. I sort of wonder how people come to postulate these things, or if they just sort of stumble on it.

Section 3.1.2, due on October 5

I found it rather difficult to understand the reimann zeta function. It just did not quite click for me how they turned the sum into the product. I need to work on that proof a bit more.

Something I found interesting was how it is possible to set an upper and lower bound on the number of primes to a particular number. I think this is similar to the reimann hypothesis, but I could be way off.

Exam Review, due September 30

I feel that the most important theorems are that a^phi(n) = 1 mod n. Also, that sum of phi(d) for all d that divides n is equal to n.

For this exam, I would expect to see problems similar to those found on the homework. I would in particular expect a problem on the Chinese remainder theorem, quadratic residues, and some theorems to prove on GCD, euclidean algorithm, and the like.

I personally need to brush up on the Chinese remainder theorem, but other than that, just a general review of all the material should suffice.

I would really like to see another Chinese remainder theorem worked out.

Section 3.1.1, due on September 28

I had a difficult time with the proof involving polynomials. I do not quite understand the logic, and perhaps more importantly how one would ever think to prove a result like that. I suppose it just takes time.

I did find it interesting that the book spent a whole section just talking about various proofs. I had never thought that various proofs of different things would be so worthwhile.

Jacobi Symbols, due on September 26, 2011

I found the Jacobi symbols to be about the same as the Legendre symbol. However, I did not quite understand the distiction. I take it that the Jacobi symbol is just a generalization, but I do not quite understand what it is generalizing to.

From this, I think that we will just keep generalizing things to more and more abstract areas. I suppose it give more power in definining things, but sometimes, it is hard to see the big picture.

Section 2.6, due on September 22

I did not quite understand the proof in regards to the Gauss sum. I did not quite follow how it all worked together. I understood where the p-1 / 2 was coming from, but not much past that.

However, the geometric proof I thought was quite good. It made the Gauss sum make more sense in my mind. The alternate proof shed some good light on the subject, sort of like how the alternate proof of the summing of the phi function in regards to cyclic generators shed light on what was happening.

Section 2.5.2, due on September 20

I had a difficult time seeing how the quadradic formula works for mod n. It intuitively makes sense, but it seems like it would not be well defined because the value under the radical could be manipulated to different values mod n. However, I suppose it is well defined, but I am looking forward to learning more about it.

Now, changing the various values mod n for example, x^2 + x + 1 mod 3 to x^2 +4x + 4 was incredible to me. Something that looks like it should have no solution turns out to have one. That was quite cool.

Section 2.5.1, due on September 17

For me, it was rather difficult to piece together the Chinese remainder theorem. It took me a little while to understand, and I still do not quite get how it all works together. The part that I am most struggling with is the a^(-1)b as being a solution. I suppose that it has to do that you are trying to find something congruent to this? I understood the method better through the examples, but still not the theory.

I did like the method for solving the ax+b equations. I remember that we covered that a little bit in Cryptography class, so that was nothing too new. I feel that I understand that part all right.

Section 2.4.5, due on September 15

I did not quite understand the proof involving the division algorithm. I had a very difficult time following it. Intuitively, it makes sense that every subgroup of a cyclic subgroup is cyclic, but the proof was difficult to understand.

I like how the phi function just keeps summing to 'n'. It seems like it is a very useful function from the properties it holds.

Section 2.4.4, due on September 12

I found it rather confusing how cyclical groups played into the whole equation. Basically, I just found them to be confusing. In general, the proofs were a little difficult to follow. I think I just lack the understanding from the previous section.

However, I did find it interesting about the ability to raise a to the phi(n). That was something that I was not completely expecting. Overall, it seems that congruence to 1 is rather important.

Section 2.4.3, due on September 10

I had a very difficult time understand the the phi function. I am not quite sure what it is intended to do. In other words, I cannot really see where we are going by learning this particular piece.

However, I think I have heard of something like this function in the past that allows one to encrypt. I think it went something like raising to a power, and then raising that integer to another power mod something to get it back to the original. I guess I will find out in a bit.

Section 2.4.2, due on September 8

At first, I did not understand Wilson's theorem. I did not think that it was proven that each identity had a unique inverse and vice verse. That sort of threw me off. Other than that, it all made sense.

The most interesting part for me was discovering that Zn is a field when n is prime. That still sort of blows my mind because they are integers. There is the well ordering principle, yet it is also a field.

Sections 2.4 through 2.4.1, due on September 6

I found the most difficult part of the material to be to get all of the definitions matched up with what I all ready know. For example, I have never heard of a "Residue Class", I have always just called them equivalence classes. Getting all of the terminology straight was the most difficult.

I like how they took the general ideas from concepts such as modulo, and built them out of things which we have all ready covered such as the division algorithm.

Section 2.3, due on September 1

For this section, I found the last part, about UFC's to be a little confusing. I think it was because I wanted to relate to the rationals. However, UFC's do not make much sense in that region. I think more practice and exposure to units and associates would help the most. Working through some examples would also be helpful.

The fundamental theorem of arithmetic was interesting when extended to the rationals. I had not thought of the set of rationals to be defined as integers raised to integers. In that way, it makes quite a bit of sense. It helped make the cardinality of the rationals make a little more sense in connection to the integers.

Sections 2.1 and 2.2, due on August 31

The most difficult part was the idea of ordering and the induction principle. I did not quite grasp what the induction principle actually means. Is it say that all subsets of N can be mapped to N? Basically, how to use this principle in conducting a proof.

The discussion of the GCD and LCM, and especially how their product is ab startled me. I had never really thought if it that way before. Furthermore, while I have heard of the GCD being the least element of the set of linear combinations, it is still not intuitive to think of it in that manner.

Introduction, due on September 1

I am a senior majoring in mathematics. I hope to become an actuary after I graduate. I have taken analysis 1,2 (341,342) Fundamentals of mathematics (290), Differential equations (334) Linear algebra, and multivariable calculus (313,314).

I have always been interested in number theory since I found out what it was. I want to learn what it is, and get enough experience to see if I would like to make a profession out of a related field.

Dr. Doud was the most effective. He was always through in his explanation of principles and concepts. He helped me understand the 'why' of the subject, and not just the how.

Dr. Li was the least effective. Difficult to understand, and I did not feel like I was learning so much as just memorizing for tests.

I count myself unique in my enjoyment of swing dancing.

Office hours work (Well, I am free from 2-3).