Section 3.3 Part 3

Difficult: Especially on pages 118-119, when the sums are changing form over and over I get lost. I admit that I have a hard time fully follwing this part of the proofs. When we talk about them in class I can mainly see it, but not always, and that is what was the hardest in the proof.

Reflective: Theorem 3.3.7 seems very interesting to me. I would have never thought that there would be arithmetic progressions of length k. If possible I would love to see the proof some time during the class.

Section 3.3 Part 2

Difficult: I find that I have a hard time following the part of the proof, Lemma 3.3.8, where they go from the Dirichlet L-series to the Euler product representation. I don't get how the multiplication goes to summation. Does this relate back to something we did with Riemann zeta function?

Reflective: I still find it amazing that mathematicians thought up these proofs. They are so layered and complex, that it just blows my mind.

Section 3.6

Difficult: I didn't think there was much that was that hard to understand in this section. The math is all very simple, and for once I think the book actually described it rather well.

Reflective: I think it is interesting that we have these relations between various arithmetic functions. I am interested to see how this all plays out in the next section.

Section 3.3 First Half

Difficult: I didn't know what a root of unity was, so that made it harder to understand the proof. I looked it up and now can follow it, but it doesn't come the most intuitively.

Reflective: I think it is interesting that we have so much knowledge about the characters mod k. What I find the most interesting is how each proof is very straightforward and simple, nothing really that complex.

Section 3.2.5

Difficult: I got lost when they used Lemma 3.2.5.1 in their proof of the infinite primes. I guess that probably comes from the fact that I didn't understand the Lemma that clearly when I read it in the first place.

Reflective: Another proof. I am not sure what this tells us about the primes though, but I am sure that it adds something of value.

Section 3.2.4

Difficult: I find myself a little bit lost around the part of the proof here we show the existence of m'. I think I mostly follow it, but it is the most hazy part.

Reflective: I think it is interesting that we know the fewest number of terms to the k power we need to get every other term. This may not be very applicable, but to me it is a very interesting fact.

Section 3.2.2

Difficult: I got lost during the explanation of Fermat's two-square theorem. I could understand the first part, but when the began to add all the different functions I got lost as to what they were speaking about.

Reflective: I think it is very interesting that we have defined all the numbers that can be the sum of squares. I am also interested to see how the Dirichlet character and function will be used later.

Section 3.2-3.2.1

Difficult: The infinite descent proof confused me. Someone the transitions seem hand wavy to me. I can mostly follow the in between steps, but it doesn't come together as I was expecting.

Reflective: I don't really know what to say. I am interested to see where this is going, but I don't yet see why this is going to apply to the infinitude of primes.

Section 3.1.5

Difficult: I got lost on the proof of Lemma 3.1.5.5. It just keeps going and I don't think I get the very first part quite well enough to follow it through. I also don't understand why in Lemma 3.1.5.1 we need primes that are congruent to 1 mod 4 in the first part.

Reflective: It is interesting to see some proofs of specific cases, but I would rather just get straight into seeing the main proof.

Rest of Section 1.3.4

Difficult: Some of the proofs of the Fibonacci numbers leave me confused. They treat some of the inductive steps of the Fibonacci numbers as obvious, but I am not seeing them.

Reflective: It was interesting to see all the various properties of the Fibonacci numbers. Some seem pretty obvious, but others were pretty crazy. I am surprised that there is so much interesting about them.

Section 3.1.4 through Page 72

Difficult: The most difficult part was just remembering all the various geometric rules. I havn't dealt with many of them in a long time, so I was rather hazy on some of them. Other than that it was pretty interesting.

Reflective: I found it rather cool to see how easy it was to draw a line with length equal to the golden number, even though it is irrational.

Section 3.1.3

Difficult: I got lost on the proof of Euler's theorem on the relation of perfect numbers to Mersenne primes. I don't understand how we can directly state the value of the sum of the factors of u.

Reflective: I didn't know that perfect numbers had any way to directly calculate them. I guess they still don't since Mersenne primes don't either, but it is still interesting to see this relationship. I wonder what this means in a larger picture sense.

Section 3.1.2

Difficult: All of it was pretty hard to follow, but the worst was the Riemann zeta function. I just had a hard time on the second page of its description.

Reflective: Analytic proofs are an interesting difference to most of the proofs we have been doing, but I have to admit they are much more confusing for me.