List of First 10,000 Primes
http://www.isthe.com/chongo/tech/math/prime/mersenne.html#M43112609
The prime number discussed in the following video was the largest known as of January 25, 2013. Since then three more larger primes have been discovered, see updates below the video.
Watch from 1:30 to 12:45
UPDATE - The newest, largest Mersenne prime as of October 10, 2017 was discovered on January 7 of 2016. It can be found at this link: http://www.mersenne.org/primes/?press=M74207281
This find was announced also in the New York Times (which can be a little more fun to read than a math site). That article can be found here http://www.nytimes.com/2016/01/22/science/new-biggest-prime-number-mersenne-primes.html?_r=0
Also, here's a video about that one:
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NEWER UPDATE - The above (and one other) was surpassed on December, 21 2018:
The largest known prime number (as of December 21, 2019) is 282,589,933 − 1, a number with 24,862,048 digits. It was found by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018
NPR Announcement: https://www.npr.org/2018/12/21/679207604/the-world-has-a-new-largest-known-prime-number
