Monday, January 01, 2007

A perfect cube

For those born in 1979 (and that includes me and a great many of my friends) last year was a big one. We're 27, or 3×3×3, or 33, and it's probably the last time we can count our years in the form nn, although let's hope not the last perfect cube.

Speaking of which, it's so easy to forget how low an odometer can read:

It was a major year for me in a number of other ways as well. About a month after adding three letters to my name, I bought my first new car. It has many more cubes to go–in fact, its first scheduled maintenance isn't until after 463 miles.

But back to the number at hand. Our time spent being 27 is split between the year just past and the year just begun, and as has been the case for the past seven years, our age bears some resemblance to the number at the top of the calendar; just drop the zeros, and there it is. But this year there's another connection, albeit one that is purely numerological and not at all mathematical.

When I took a course in algebra from K. Ribet, he mentioned an algebraist who every year published a list of all the groups with order equal to the year. One of my classmates pointed out that in certain years, this would be a rather short list; for instance, in 1979 only one group qualified. This year isn't as dull as all that, but it is manageable, since 2007=3×3×223. For instance, I know from one of Prof. Ribet's homework assignments (specifically, Problem 28) that all such groups are solvable and have at least one normal Sylow subgroup.

But what I like, even though I know it's nothing but a quirk of our base-10 notation, is the typographical similarity between the prime factorizations of 27 and 2007, 3×3×3 and 3×3×223. That the products differ by two zeros and the factorizations differ by two appearances of the numeral 2 makes the numerological aesthetics all the more appealing. I know it's not math, but it sure is pretty.

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