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gjm comments on Open thread, Feb. 9 - Feb. 15, 2015 - Less Wrong Discussion
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More like 25, actually; the first paper on these numbers was published in 1990, long before Colton's program. So far as I can tell, in those 25 years no one has found anything else (in pure mathematics or elsewhere) that is made easier, or understood better, as a result of having the notion of "refactorable numbers" available. So perhaps rather than "too early to say", I should say that they've been discovered and found to be nothing more than a curiosity.
But I don't think I should. In mathematics, sometimes quite a long time passes between when a thing is first thought of and when it's first found to be useful. Maybe refactorable numbers will turn out to be a key concept in the proof of the Riemann Hypothesis in 2137. I wouldn't bet on it, though.
Lest I be misunderstood, that doesn't mean I think there's anything wrong with being interested in refactorable numbers or studying their properties for their own sake. But if they're just a curiosity, as seems to be the case, then I'm not so impressed by HR's achievement in finding them. It seems like one could do about equally well by picking (say) ten simply-defined properties of a positive integer (say: number of divisors, sum of divisors, sum of squares of divisors, number of prime factors, sum of prime factors, Euler totient function, number of 1s in binary expansion, Ramanujan tau function, floor of square root, floor of base-2 logarithm), considering all pairs of them and putting a few things like "equals", "divides", "equals square of" in between. I bet that at least 10% of the results will define sets of positive integers that are neither much less new, nor much less interesting, than the refactorable numbers were when HR found them.
Can you give an example of something in mathematics that was invented, condemned as boring, and a decade or more later found to be useful?
Embarrassingly: no, not offhand. I have a general impression that this happens sometimes, rather than specific examples. Maybe some of the specialfunctionology used by Louis de Branges to prove the Bieberbach conjecture?
Supposedly it did discover some interesting things from it. Like if the sum of the divisors of an integer is prime, then the number of divisors must be prime. That seems interesting, but I'm not a mathematician.
Well, let's see. Sum of divisors is product of 1+p+...+p^k so if that's prime then the number must be a prime power so only one p, and then if the number of terms in that sum is composite you get an obvious nontrivial factor for the sum of divisors, and we're done. I reckon something I can prove in two minutes without making use of the notion of refactorable numbers probably isn't a great argument for the importance of that notion.
Perhaps the point is that this theorem was discovered while thinking about refactorable numbers. I rather doubt it, but in any case the theorem itself seems like a cute curiosity rather than something any number theorist would care much about.
(I am a mathematician, though I've been out of academia for years and was never a number theorist.)