As of March 7, 2013 the largest prime is 2^57885161  1 discovery January 25 2013. It even made it to the news, http://www.cbsnews.com " Largest known prime number is 17 million digits long". If anyone is interested to see what it looks like written in English  http://www.isthe.com/chongo/tech/math/digit/m57885161/prime.html or using digits  http://www.isthe.com/chongo/tech/math/digit/m57885161/primec.html
Lenim Depina
I really enjoyed reading that page that Lenim posted where the largest prime was written in English. I had to start at the bottom first though and work my way up, though, so I could understand it better.
We have been discussing a number of topics in number theory recently, and fortunately, technology has made it possible for a large amounts of mathematical theory to be computationally proven, and the formation of algorithms that are implementable by computers, it is quite easy to make such discoveries without so much the worries of the calculations being computationally laborious.
One example is this interesting page I found on Rutgers' website, that lists (or at least claims to, I have not confirmed) 1098 primative Pythagorean triples. Remember, a primative Pythagorean triple is such that at least one of the three values is a prime number. While some, such as 3, 4, 5 containing two primes, obviously the distance between primes increases as numbers become larger, with more numbers having larger divisors that are difficult to determine sometimes without computers, and other triples such as 7 24, 25 have two composite (nonprime) values.
I also wanted to investigate more about Sophie Germain. It is unfortunate that she had to pose as a man in order to be recognized by the mathematical world, as there are many intelligent women that are immensely important for the advance of sciences and the humanities, none of which are made any less intelligent by the fact that they are women. PBS has an excellent biography of Sophie Germain, detailing not only the events leading up to her life, particularly Fermat's Last Theorem and challenge he posed, but also details Germain's upbringing. I learned a number of amazing things about Germain through this:
 She was only able to maintain her research through funds from her father, who was considered quite well off at the time.
 Born during the French Revolution (April 1st, 1776), Germain's education was overshadowed by the Reign of Terror and other brutal events that occurred during the revolution.
 Carl Friedrich Gauss, who the article claims was actually very welcoming of contributions of women to mathematics, especially considering how women were viewed at the time. While they corresponded for quite some time, Gauss eventually stopped returning her letters when he was given a professorship and "his interest shifted from number theory to more applied mathematics."
 Napoleon was actually very much responsible for making sure Germain's work's were attributed to her. According to the article, "Germain's contribution would have been forever wrongly attributed to the mysterious Monsieur Le Blanc were it not for the Emperor Napoleon."
Overall, I found PBS' article to be not only enjoyable to read but highly informative, providing important background information to Sophie Germain's works that helps to get inside her mind as she attempts to prove a very significant theorem in number theory.
Finally, while I have been able to calculate the GCD of two numbers using the Euclidean Algorithm, I am still working on an algorithm to simultaneously calculate ax + by = GCD(a,b). A recursive implementation seems difficult but rewarding if accomplished.
—Rob Moray
The largest prime number written out in english is interesting to see. Also I was not in class this Thursday but I found a pretty thorough explanation of the Euclidean Algorithm in this pdf: http://aleph.straylight.co.uk/eea.pdf it is helpful as a review as it explains both how to use it and why each step is valid mathematically.
Angela
The largest prime number written out in English is really interesting, I try to write it in Chinese, but I cannot because it's too huge to write and I do not know how to write it in Chinese accounting system. However, I started thinking if we can rewrite this largest number in another way? Is it too crazy if people rewrite it in Binary? I think that's the worst case. And I think if people could smaller this number in a new calendar, and provide new algorithm for the calendar, they may get next largest prime number easily….

Jun