### We were once so close to Heaven

Oct. 26th, 2009 01:13 amBut I digress.

I was playing around with transcribing my college notes into an electronic format when I came across a Math Studies Problem Seminar problem that I hadn't solved. (I had a lot on my plate that semester.) The problem was to find one or more interesting facts about antiprimes (often called "highly composite numbers", defined as numbers that have more factors than any number less than it) and to compute all antiprimes less than 100 without electronic devices. So I worked out a pretty routine algorithm that would do the latter part that needed two lemmas to justify it, and thought that those would be the interesting facts that I would prove. The first was routine (and is the first two observations about the prime decomposition of an antiprime given on the MathWorld link above if you were curious), but the second was harder. I wanted to show that if n is antiprime and greater than 1, then n has a prime factor p such that n/p is also antiprime. It seemed obvious from poring over the first few dozen antiprimes, but there wasn't an easy proof to be found. So I banged my head against that for a good long while, and then gave up and posted to xkcd's math forum to see if they could make anything of it. (Huh, in retrospect, Usenet is not only dead but the mourning period is over, because this is the first moment this weekend I've considered sci.math.) And after some other good people banged their heads for a while, damned if another of the good people there didn't find a counterexample.

3622794316246739379743037382304885029330

I know, why didn't I notice that before wasting the time of nice forum people, right? That is likely the biggest number that has ever been generated on my behalf. I suspect the hypothesis is more interesting than some boring old true one would be.