Why are perfect numbers important? What is the best way of introducing these numbers to a first course on number theory? I could not find any application apart from the relation to Mersenne primes. Are there any other applications of perfect numbers?
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2$\begingroup$ What is the specific question here? $\endgroup$– Benjamin DickmanCommented Jul 21, 2014 at 16:56
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8$\begingroup$ There is a premise of the question that seems not to be very well supported: that the perfect numbers are important enough that one has to include them in a first course on number theory. It feels difficult to answer the question without this somewhat dubious assumption. $\endgroup$– Benoît KloecknerCommented Jul 21, 2014 at 18:56
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5 Answers
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I think it turns out that "perfect" numbers do not interact much with other parts of number theory. Some of these very old, elementary, very ad-hoc definitions of special classes of integers have proven (and will prove) to interact interestingly with other ideas, but some seem not to. It's not easy for a beginner to guess the significance or subtlety of one of these classes of integers from the innocuous elementary-sounding definitions, usually. But tradition itself lends some significance to such notions. And there is always the possibility that the future discloses some surprising connections. And sometimes the sheer long-standing irresolution of a question gives it considerable cachet, even without profound consequences.
answered Jul 21, 2014 at 17:04
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2$\begingroup$ This answer is perfect, but it doesn't mention explicitly that the original definition of perfect numbers were just one of the many classes of numbers invented by the ancient Greeks. Their motivation is unknown, but was probably purely aesthetic. Maybe someone noticed that $6 = 2 + 3 + 1$ and got curious. $\endgroup$– Jack MCommented Jul 22, 2014 at 16:33
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1$\begingroup$ @JackM, ah, yes, there were/are also "amicable" numbers, "idoneal" numbers, and others I've forgotten. Some mysticism, too, I think. $\endgroup$ Commented Jul 22, 2014 at 16:35
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4$\begingroup$ IMO "idoneal" does not really fit the list; the notion is some 2000 years more recent (Euler). $\endgroup$– quidCommented Jul 24, 2014 at 10:52
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Mathematically, even perfect numbers give a good number theory example to the general idea of classification, i.e. all even perfects have a specific form.
I use perfect numbers in my number theory class for two or three pedagogical reasons: with some trial and error (and the help of some computational software), I have the students essentially discover Euclid's formula. There's a lot of good conjecturing and testing along the way. We then look at the proof which is one of the few that are long enough to be nontrivial from the student view, yet easy enough to follow that they can actually understand it (especially after the preliminary explorations). When it's all done, I talk a little about odd perfect numbers: no one knows of any nor can prove they don't exists. This ties into a common thread in my upper level courses: math is still alive, still being discovered, still full of the unknown.
All this being said, I surely wouldn't look down on a teacher or student who knew nothing about perfect numbers and didn't have them covered in class.
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3$\begingroup$ Not sure why this was downvoted. Seems like a perfectly reasonable response. $\endgroup$ Commented Jul 24, 2014 at 4:02
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$\begingroup$ @StevenGubkin, it is almost the one I was going to give... $\endgroup$– vonbrandCommented Jul 23, 2015 at 1:30
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$\begingroup$ Upvote for the mention of odd perfects alone - I spend FAR too much time on this in my own class/book, it's so tantalizing, and one of those things like P/NP where you can get experts disagreeing on what the expected answer "should" be. $\endgroup$– kcrismanCommented Mar 10, 2017 at 5:21
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I don't know of a practical application of perfect numbers per se (not saying they don't exist), but its historical status dating back to the Greeks makes them interesting for pedagogical purposes. The Greeks (the Pythagoreans in particular) put mystical significance on special sequences such as the primes, the figurate numbers (notably the triangular numbers), and of course the perfect numbers. If you find that injecting anecdotes into your teaching helps you grab the attention of your students, then perfect numbers are a well-documented example of the ongoing human fascination with number theory since antiquity.
For a very detailed history of perfect numbers, check out http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html. (In fact entire MacTutor History of Mathematics Archive is quite fascinating, and makes for quite nice leisure reading.)
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We "need" the natural numbers to count, we need the real numbers to measure and we need the complex numbers to guarantee that every quadratic equation has solutions.
Whenever we have some collection of objects and we can create a new definition that makes it possible to make distinctions between the objects (some are the "same" and some are "different") the potential exists to get new insights into the objects involved, and to other related objects.
Thus, we can talk about the integers that are prime, those that are squares, cubes, the sum of two squares, etc. Some of these definitions create wonderful new theoretical and applied playgrounds. The primes are a good example. After noticing the primes then one can show the prime factorization theorem. Primes can be used to design a cryptographical system (RSA) powerful enough to protect many financial transactions at the current time.
Perfect numbers create a "playground" for the interested. One of my undergraduate professors, Leo Zippin, made the observation to me that if some mathematician can create a "nifty" bit of theoretical mathematics, eventually some other mathematician will find a "nifty" use for those ideas.
answered Jul 21, 2014 at 20:03
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Coming late to this question ... but hopefully with something useful.
Although I find the historical reason plenty, as well as ease of proof, there is something to be said for approaching not perfect numbers, but rather the summation function/aliquot divisors function, as the key concept.
That is, if we start with the reasonable function $\tau(n)=\sum_{d\mid n}1$ and try to find patterns in it, such as when it is even or odd, one might also try to find patterns in the related $\sigma(n)=\sum_{d\mid n} d$. And also $\phi(n)$, etc. What is one pattern you might seek? Well, do you ever get a perfect (!) multiple of your input as output? (As well as whether it is multiplicative, and lots of other questions.)
So in an introductory number theory course, there is lots of room to look for interesting patterns in many arithmetic functions, and then the perfect number question is just a nice example of one we can actually answer - but surely not the only one, or even a particularly important one from this point of view. (Personally I enjoy exploring asymptotic behavior of arithmetic functions.) I like all the answers here, but this is one approach to take for motivation if for some reason you didn't like those answers.
