Keeping things in context

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I’ve started reading through Stewart and Tall’s book on algebraic number theory, partly to give myself some fodder for learning Sage and partly because it’s an area of math I’d like to explore. I’m discovering a lot about algebra in the process that I should have known already. For example, I didn’t know until reading this book that the Gaussian integers were invented to study quadratic reciprocity. For me, the Gaussian integers were always just this abstract construction that  Gauss invented evidently for his own amusement (which maybe isn’t too far off from the truth) and which exists primarily so that I would have something to do in abstract algebra class. Here are the Gaussian integers! Now, go and find which ones are units, whether this is a principal ideal domain, and so on. Isn’t this fun?

Well, yes, actually it is fun for me, but that’s because I like abstract nonsense and I like just constructing rings out of nowhere and seeing what works and what doesn’t. But this approach to algebra is a lot harder to convince others to adopt, particularly college math majors whom I teach, most of whom struggle with abstraction. For them, any connection, no matter how tenuous, to the real world is a comfort and a reason to study. Quadratic residues aren’t exactly in the same league as designing airplanes in terms of “real world” utility, but it’s at least something that’s easy enough to understand and explain. Even if you care nothing for real world utility, it’s important to know why something was invented when you are setting about studying it. Otherwise you learn a subject in abstraction and without connections to its roots.

In fact, it seems like a lot of what we take as being canonical in abstract algebra was invented to study number theory. And yet, I have never taken a number theory course, and the number theory that was included in my studies of algebra was added mainly to set up the study of abstract groups and rings, as if to say that number theory exists to make studying algebra easier instead of the other way around as appears to be the case. And it’s not because I had a bad algebra education; I studied under some of the best algebraists around, but I never got the memo that abstract algebra was for something. I learned algebra mainly in isolation for the sole purpose of calculating homotopy groups. Likewise, my entire grad school training was focused on topology, which is supposedly a branch of geometry, but the only course in geometry I have in my background was Mrs. Buttrey’s class at William James Junior High School in the eighth grade — and that didn’t exactly give me the disciplinary perspective I needed to put topology in its proper context. (Even though it was a really good geometry class — thanks Mrs. B!)

I’ve been thinking that my post about the, er, pedagogically challenged way that Stewart Calculus does its examples about instantaneous velocity is really about the idea that you need to make sure that a person learning a new idea has some reason to learn it, before you give it to them in full complexity. Or at least before they’ve finished a course in it. Perhaps this idea extends to all of mathematics and maybe even beyond.

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Undergraduates make a “prime” discovery

This is cool:

Westfield State College senior mathematics majors Jeffrey P. Vanasse and Michael E. Guenette, working under the direction of Mathematics Department faculty members Marcus Jaiclin and Julian F. Fleron, have made a significant new discovery in the mathematical field of number theory. They have discovered the first known example of a 3 by 3 by 3 generalized arithmetic progression (GAP).

Most easily thought of as a 3 by 3 by 3 cube (similar to a Rubik’s cube puzzle) made up of 27 primes, their discovery begins with 929 as its smallest prime ends with 27917 as its largest prime. The intervening 25 primes are constructed by adding combinations of the numbers 2904, 3150, and 7440 in an appropriately structured method.

“Such an object was known to exist and its approximate size had been loosely estimated,” Fleron said. “However, a blind search would require checking more cases than can be feasibly checked by all existing modern computers each running for the next million years. Instead, the group used knowledge of the structural relationships between the potential candidates to greatly reduce the potential candidates to be checked.”

Whole thing here. It goes to show that students and faculty can work together to produce some really first-rate mathematics, if they’re put in a position to have the time and materials to make it happen and in an environment that really values that kind of scholarship. It also shows just how great of a role computing, especially computer programming, plays in doing mathematical research these days. Congrats to Jeffery, Michael, and their profs for a job well done.

It’s a prime! And another prime!

The number believed to be the 45th Mersenne prime has turned out actually to be a prime, according to GIMPS. The verification was completed on 6 September and announced on 7 September.

But in a fairly extraordinary turn of events, yet another number was submitted to the GIMPS servers as the next possible Mersenne prime on 6 September — and the initial verification shows that it is prime too! So we now have the 45th and 46th Mersenne primes discovered within two weeks of each other, which is incredible.

No word yet on the details of these primes. We’ll soon see who wins the Mersenne prime digit-guessing challenge. You can still play along with your own spreadsheet too!

Estimating the digits in a Mersenne prime — for dummies

At the end of this post, I made a totally naive guess that the recently discovered candidate to be the , the 45th Mersenne prime, would have 10.5 million digits. There was absolutely no systematic basis for that guess, but I did suggest having an office pool for the number of digits, so what I lack in mathematical sophistication is made up for by my instinct for good nerd party games. On the other hand, Isabel at God Plays Dice predicted 14.5 million digits based on a number theoretic argument. Since I am merely a wannabe number theorist, I can’t compete with that sort of thing. But I can make up a mean Excel spreadsheet, so I figured I’d do a little data plotting and see what happened.

If you make a plot of the number of digits in , the nth Mersenne prime, going all the way back to antiquity, here’s what you get:

The horizontal axis is n and the vertical axis is the number of digits in .

Admit it — one look at this plot and you’re itching to add some trendlines. Here’s what you get when you add both an exponential trendline (perhaps the obvious choice given the shape) and a 6th-degree polynomial:

The exponential one has a higher value, but that’s perhaps misleading because of the really good fit for all those low-digit Mersenne primes that happened prior to around . We’ll take that issue up in a moment. But for now, let’s put those trendline equations to work. The exponential trendline would predict that would have a digit count of

which is obviously rather a lot more than either my prediction or Isabel’s; and if you put in into the 6th-degree polynomial, you get a digit count of 11819349, which is in the ballpark of both my rough estimate and Isabel’s estimate.

It doesn’t make much sense, though, to include all Mersenne primes, since Mersenne primes didn’t even cross the 100-digit mark until in 1952. A more accurate idea — if you can call this kind of reasoning accurate in the first place — would be to run the numbers starting at around and seeing what we get. I’ll save that for later, unless somebody wants to beat me to it.

New Mersenne prime discovered?

GIMPS is reporting that on 23 August a new Mersenne prime was reported to their server. Verification began today and should take about two weeks to complete. No word on what the prime was, how many digits, etc.

The last Mersenne prime discovered was , back in 2006 (blogged about here) and weighed in at a whopping 9,808,358 digits. Any bets on how big this new one is, if it’s really a prime? I’m guessing 10.5 million digits. Sounds like a good occasion for a nerd office pool.

Update: Isabel at God Plays Dice likes 14.5 million digits instead, and she’s actually using math and stuff to make that estimate instead of just shooting totally in the dark like I am.

(Super-)Powers of 2

Driving in to work this morning, I suddenly felt my vision go blurry to the point where I literally couldn’t see anything. Fortunately, I was able to pull off the road into the parking lot of a small office building before causing an accident. After I stopped and waited for the blurriness to subside, the first thing I saw was the mailbox for the office building, which had a street number of: 2048. Rather than wonder what the crap was wrong with my eyesight, or frantically try to decide whether to go see a doctor on the spot, instead the first thing I thought was hey: That’s . 

Then, after making it to work with no more blurry vision attacks, I walked up to my office — the same office I have been entering and exiting since summer 2001 — and looked at the office number and saw it: room 128. Of course, I’ve never had a problem remembering my office number But for the first time in seven years, I noticed, hey: that’s . 

So, maybe the blurry vision attack was me suddenly gaining the superhero power of being able to recognize powers of 2 with lightning quickness. If so, I somehow don’t see the Justice League of America saving me a seat anytime soon.