I’m just curious if it’s common for numbers to have a ton of arbitrary mathematical properties. 42, for instance, seems to have a laundry list of not-really-bizarre-but-kind-of-cool mathematical properties. Often with impressive sounding names like “pronic number” (product of two successive integers).
But I really wonder, if I picked a random integer, could a sufficiently bored mathematician find a comprehensive list of a bunch of impressive sounding but relatively meaningless properties? It seems like you could, but I have heard some say things like how certain important numbers (like 666) are actually “really boring” mathematically. However, even if some numbers are “boring” can you get a bunch of impressive sounding properties for most of them (yes, I realize the futility of asking this question about an infinite set)?
I suspect non-primes are going to fare better than primes in this department, simply because being prime excludes you from a whole class of properties involving your factors, but who knows, they may make up for it in other ways.
Claim: There is no such thing as an uninteresting natural number.
Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting. Due to the well-ordered property of the natural numbers, there must be some smallest number in the set of uninteresting numbers. Being the smallest number of a set one may consider uninteresting makes that number interesting after all: a contradiction.
Hmm. I’ve never found that proof all that convincing; it smacks of the unexpected hanging paradox to me.
But I guess that what we mean by interesting number is not a clear concept anyway, so it’s moot.
Which makes it pretty much exactly the same as the “unexpected hanging” paradox – since that one essentially hinges on the unclearness of what “unexpected” means.
There are people who really do get devoted to finding out, listing, and remembering the facts about numbers. Case in point, one of my favorite math books, the Penguin Dictionary of Curious and Interesting Numbers:
Second case in point, the story of G.H. Hardy and Srinivasa Ramanujan:
91 has two interesting properties. It is the smallest positive integers that is the sum of two cubes in two different ways. And it is smallest composite number that most people think is prime.
I remember reading that taxicab 1729 anecdote and thinking, “Seriously?” I thought some of my friends were geeky but couldn’t imagine any of them complaining to an ill friend in hospital that the number of their cab was “dull”!
This is oozing into the metaphysical, but the Kabbalistic (new age version, don’t know about orthodox version) of reality is that ANYTHING can, and should, be connected to something else in an interesting way. Numbers, words, people, objects, concepts, this pen, that tree…connect everything there is to connect, and the last things left to connect are you and YHVH (God.)
