Wikipedia:Reference desk/Archives/Mathematics/2010 April 20

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April 20

Distribution of size of largest prime factor

Resolved

I'm sure this is available somewhere in this encyclopedia but not exactly where. The question is what the probability is for a randomly chosen large positive integer that its largest prime factor is in some range. However that's stated wherever it is is fine with me. I'm trying to determine the expected number of bases, B, less than some large N s.t. the largest prime factor of 1223334444...(B-1)(B-1)...(B-1) is an emirp (B=10 is the only solution for B<17).Julzes (talk) 04:25, 20 April 2010 (UTC)

[1] gets a bunch of hits. 66.127.54.238 (talk) 07:30, 20 April 2010 (UTC)

Thanks. I'm just marking this as resolved. It seemed like a silly question when I asked it, it seems more so now, and after I have breakfast tomorrow I'll probably find it to the degree desired in my Hardy and Wright without even checking the Google results.Julzes (talk) 07:43, 20 April 2010 (UTC)

Math contests

Is performance on math contests (AMC, AIME, USAMO, etc.) a good indication of mathematical ability in general? --71.144.122.18 (talk) 23:56, 20 April 2010 (UTC)

Mathematical ability in general is sort of broad. The skills involved in contest math can definitely be called a subset of it. If you mean more specifically is it indicative of ability in like proof-based higher math, then I'd say some of the skills probably are and some aren't. Contest math puts a premium on speed, knowing certain formulas by heart, doing arithmetic efficiently and accurately, and making good guesses, which are skills that are not so important for doing proofs. But the insight needed to find the "trick" in contest problems and being able to carry principles across different branches of math I think is similar to the insight needed in proving results. And more generally having a good head for understanding math concepts is important for both. Just anecdotally, most of the people I know who were good at contest math in high school probably could have been successful in pursuing math further if they'd wanted to, but I could be misjudging that. Rckrone (talk) 01:15, 21 April 2010 (UTC)

As Rckrone says, mathematics (and, in particular, mathematical ability) is very broad. No mathematics contest, especially one at the school level, can capture every technique and intuition in formal mathematics. In fact, math contests in general tend to focus on the basic aspects of certain areas of mathematics, such as number theory, combinatorics, and Euclidean geometry (at least in my experience). There is usually little or no emphasis put on "infinite mathematics", branches of mathematics such as modern algebra, analysis, topology etc., and these are really diverse areas of mathematics (of course, this is mostly due to the fact that the contests are aimed at high school students). That is not to say that number theory and combinatorics are not diverse; they certainly are prominent areas in the mathematical world. However, within these math contests, even the breadth of the number theory and combinatorics tested is quite narrow (as I said, because these are high school contests, a very limited number of concepts can be tested). In number theory, for instance, there are many important areas such as algebraic number theory and analytic number theory, whose techniques and intuitions do not lie within math contests in general.

All that said, it is definitely the case that someone who can do well in math competitions is intelligent. However, there is so much more to mathematical intelligence than what lies in math competitions. In that regard, I disagree that students who do well in math competitions would definitely become good mathematicians (another important quality which comes to mind, that competitions do not test, is patience; the greatest mathematicians of all are also the most patient ones, in my opinion). Conversely, someone who cannot do well on math competitions may well go on to become a great mathematician. Succinctly, doing well in math competitions, and doing formal mathematics are almost completely independent of each other; it is extremely difficult to decide how good a mathematician a student will become based on his performances in math competitions (and by "good", I mean the student's potential; he may not become a mathematician later on, but even if that is not the case, it is still difficult to decide his potential). PST 02:08, 21 April 2010 (UTC)

I think that in line with the notion that one needs patience to be a top-flight mathematician, as opposed to a good competitive mathematics student, is a discernment for the difficulty of problems that have no known solutions yet. An expectation can be raised in competitions that whatever problem you work on will fall before you if you consider it well enough. In actuality, a first rate mathematician is going to have to be reasonably selective in what he or she spends time on and with the hardest problems is apt to just give up--perhaps returning later in life--when the ideas aren't forthcoming. However, good contest mathematics differs little in kind from research mathematics. Aside from the fact that the competitions generally end at the undergraduate level, leaving only such things as posed problems in journals as similar, the main difference between the one kind of mathematics and the other is generally a real divide between the certainty of the existence of or the amount of time required for a solution.

As far as poor contest participants being good mathematicians in the end, I think there are certain factors at play like the age at which one decides finally to focus on mathematics and choices in favor of depth rather than solidity (studying advanced topics when skills are still middling for the more elementary ones). I don't think there is a solid mathematician who couldn't train him- or herself to be good at the contests; and I suspect that for the vast majority who have become recognizably good mathematicians very little time would be required for it, but there wouldn't be any point for their research, with only possible benefits as regards their pedagogical tasks.Julzes (talk) 03:12, 21 April 2010 (UTC)

I should add/note that library skills for the profession are important for getting anywhere in mathematics, like other academic fields. A professional mathematician has to sort out what is already known, not only fully but partially. Unless one is working on highly idiosyncratic problems, working with the literature is something one must do that is nearly totally unnecessary not only for contest mathematics but mathematics pre-doctorate.Julzes (talk) 03:25, 21 April 2010 (UTC)