Open thread, Feb. 9 - Feb. 15, 2015

MrMind

If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

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If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

Notes for future OT posters:

1. Please add the 'open_thread' tag.

2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)

3. Open Threads should be posted in Discussion, and not Main.

4. Open Threads should start on Monday, and end on Sunday.

Too early? It was discovered 16 years ago. Now it has it's own Wikipedia article and some other hits, so someone must have found it interesting.

But yes this is the biggest problem with automated discovery, that there is no definition of "interesting". Automated discovery systems tend to produce random garbage after enough time. One paper defined it as things which are difficult to prove, and so it discards trivial and obvious stuff (EDIT: better paper on this subject.)

Speaking as a mathematician who wrote one of the papers cited there, the concept isn't very interesting at all. It is the sort of recreational mathematics that we enjoy playing with and is fairly natural but it isn't the sort of thing that is going to lead to deep insights or structural mathematics. The bar of being interesting enough to have papers written on it is really low.

4gjm11y

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.