Today, I noticed both Zachary Abel and Qiaochu Yuan plan to write a blog post every day this month (hooray!). I haven’t written on here as much as I would like recently, and so I thought, why not? I already missed November 1, but no matter, perhaps I can do a bonus post on December 1 to make up for it. So, I hereby commit to write one blog post every day this month!¹

So what will I write about? Well, a long time ago I wrote about the Lucas-Lehmer test for finding Mersenne primes, and promised to prove it in some future posts. I never got around to that, of course, and it seems like a perfect topic. I don’t know if we’ll make it through the whole proof, but it doesn’t really matter. The posts will be short, and we can take things slowly and explore interesting things that come up along the way. Don’t fret if you don’t remember what Mersenne primes or the Lucas-Lehmer test are, I’ll explain those again as we go. And, of course, you can help shape the direction I take by leaving questions and comments.

For today, just recall what a Mersenne prime is: a prime number of the form . For example, is a Mersenne prime (the third, in fact). Can you find the first two? What is the next one?