Happy Hyperpyramidal Year!
For most people, numerology is the belief that numbers are “lucky” or “unlucky.” Many Western hotels omit the thirteenth floor, and the thirteenth room on each remaining floor, in deference to such superstition. In some Far Eastern cultures the digit four is similarly unlucky (due to a homonym with words denoting “death”.) I’ve been in hotels that try to satisfy both sets of guests… the elevator controls look distinctly gappy! A few more such aversions and the hotel would be reduced to telling every guest “your room’s up a few floors and along the hall.”
Among mathematicians, the word is used humorously for the belief that some numbers are “interesting,” especially when stated in isolation from any heavyweight theory. The Pythagoreans and their successors studied square, triangular, perfect, and other types of numbers. They don’t seem to have done so in the belief that (say) pentagonal numbers were sacred to Aphrodite, or that you should not buy fish on a perfect-numbered day. Nor, at first, did anybody study any deep theory behind these numbers that we know of. They were justified by the numbers’ beauty.
But of course, one person’s frivolous observation is another person’s PhD thesis or Fields medal. You may think it’s entertaining that 1^1 + 2^2 + \cdots + 24^2 = 70^2, and it certainly is. But Lucas conjectured in 1875 that this was the {only} nontrivial case in which a pyramidal number was also square… and it was 43 years before G.N. Watson proved it, via a very nonfrivolous utilization of elliptic functions. Furthermore, there are deep connections between this and the very high symmetry of the 24-dimensional Leech lattice, which in turn relates to properties of the sporadic simple groups.
Speaking of sums of powers, you probably remember that G. H. Hardy had a story about a time when he was visiting Ramanujan in hospital when the latter was ill and depressed, and remarked, to try to get a conversation going, that the taxicab he’d taken to the hospital had the rather uninteresting plate number 1729. Ramanujan immediately identified it (supposedly to Hardy’s surprise) as the first number that was the sum of two cubes in two distinct ways.
I’ve sometimes wondered about the surprise. Of course, Hardy knew that Ramanujan loved numbers, but nobody has elsewhere suggested that taxi numbers had a special place in his affections, as locomotive numbers do for trainspotters. And if not, why should Hardy bother remembering a taxi number for him — unless he too saw that it was interesting? And surely he could have seen it, due to the happy coincidence that, to anybody knowing the first few cubes, the partitions 1000+729 and 1728+1 are both obvious at a glance in decimal notation. I think Hardy probably knew exactly what he was bringing to entertain the invalid!
Anyhow, this is the year 2025 of the Common Era, and 2025 is a very interesting number indeed. It is (as I’ve already seen in an email from one math organization) a perfect square: the only such year in most of our lifetimes. But that’s not all! Specifically, it is 45^2, and 45 is (yes!) a triangular number. And we all remember (from first-year calculus) what the squares of triangular number are: they’re the hyperpyramidal numbers, the sums of the first n cubes! 2025 = 1^3 + 2^3 + \cdots+ 9^3. We haven’t had a year number like that since before Chaucer was born, and the next won’t be until 3025.
Add 3025 to this year’s number, of course, and you get 5050… which we all recognize as the answer that young Gauss supposedly obtained almost instantly when his schoolmaster tried to keep the kids quiet by making them compute 1+2+\cdots+100. A coincidence? Not really! If we represent 1+2+\cdots+n by \Delta(n), then it’s very easily shown that \Delta(n^2) = \Delta^2(n-1) + \Delta^2(n). Unfortunately, this seems to be as far as it goes: it’s not, as far as I can see, the beginning of a nice pattern for higher powers. Just the Strong Law of Low-Degree Polynomials at work?
I doubt if this will make the year specially lucky… but at least it’s something to think about when the news gets too depressing. I wish the best of luck to all our readers in these potentially difficult times.
