Extra Material
Lecture 27: Pythagorean Triples
Today's lecture was about Pythagorean triples.
Lecture 28: Math in Pop Culture
We started class by introducing the play Proof. Proof, written by David Auburn and published in 2001, deals with three mathematicians: a mathematics Professor, his daughter, and his former student. The class read two scenes aloud that gave way to our first two topics. The first scene was an argument between the Professor and his daughter about how many days she has wasted. After some back and forth, they realize that she has wasted a little over 33 days, which, if treated as years, and then converted to weeks, equals 1729 weeks. This number is special because it is expressible as the sum of two cubes in two different ways. Wael showed us a theorem showing how to characterize numbers that are the sums of two cubes. The second scene from the play that the class read aloud was about the daughter and the former student discussing mathematics conferences. The daughter tells Sophie Germain's story, which leads into our second theorem below presented by Jen, regarding when Germain primes occur. Afterwards, Tom discussed a conversation from the television show Bones in which the topic of perfect numbers came up. He followed with some brief history of perfect numbers, by proving and disproving some claims by Nicomachus, and following up with a neat characteristic of perfect numbers.
Lecture 29: Goldbach's Conjecture
Today we introduced Goldbach's Conjecture and how it has been modified and verified to varying degrees since its inception in 1742. This includes the strong and weak versions of the conjecture. After this we discussed the different mathematicians who had worked on the conjecture and their work. Lastly, we explained through a heuristic argument why you should believe Goldbach's Conjecture to be true.
Lecture 30: Continued Fractions
As we are all aware, there are plenty of numbers besides just the integers. Now is the time to bring this fact to the forefront. In this section, we will begin to explore the idea of continued fractions. Continue fractions are useful in many advanced areas of number theory.
Lecture 31: Hensel's Lemma
Today we talked about Hensel's Lemma.
Lecture 32: Integers as Sums of Squares
Today we talked about writing integers as sums of squares.
Lecture 33: Fermat's Last Theorem
Today we talked about Fermat's Last Theorem.
Lecture 34: Cryptography and RSA
Cryptography is used to encode and decode information as a means of security. It’s used to hide messages from non-intended recipients by making the messages unintelligible to them. In Greek, cryptography means “hidden secret.” In the past, cryptography was utilized mainly by the military and political sectors of intelligence, but presently it is commonly used in the commercial sectors for electronic commerce, computer passwords, ATM cards, and other applications. Over the years there have been many different methods of cryptography, but today the most commonly used algorithm for encryption is RSA.
Lecture 35: Partitions and the Partition function
Today we talked about the partition function.
Lecture 36: The Riemann Hypothesis
Today we provided an overview to the Riemann Zeta Function. We started by discussing the man of the hour — Bernhard Riemann. We then defined the Riemann Zeta function and performed a calculation of $\zeta(2)$, which is known as the Basel problem. We then proved the Euler Product Formula and described its importance in relative primality of integers. We also provided a few equations which relate the Zeta Function to other functions we discussed throughout the class. Finally, we concluded with a very important hypothesis: The Riemann Hypothesis.