The Ramanujan’s quote, “‘No,’ he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.” isn’t quite right or he was wrong.
He should have said the smallest positive number expressible as the sum of two positive cubes in tow different ways.
If their friend was a mathematician famous for doing fun and interesting things with numbers, they might spend the entire cab ride there trying to find a number for their friend to have some fun with.
If a number were completely uninteresting would that fact alone make it… interesting?
Only if it was the only one 
For several years, I have enjoyed Futilty Closet, some smart dude’s engaging blog about all sorts of miscellany. Many of his pieces are about numbers with interesting properties. I had started to wonder the same thing the OP did.
Here’s that same site, but just the “science and math” entries.
No doubt this is one of the reasons that numerology is popular as it is.
I thought you’d never ask.
6^3 + (-5)^3. Note that I said the smallest positive number that is the sum of two cubes in two different ways. Didn’t say the sum of two positive cubes.
Multiply by 8 to get 12^3 + (-10)^3 = 8^3 + 6^3 and it is also 9^3 + (-1)^3 so that 728 is the sum of two cubes in three different ways.
Yep got it two minutes after I posted. In my defense it was late at night and my mental subtraction wasn’t playing the game.
Still, I am not happy with the solution. Allowing negative integers disturbs the notion of smallest. I could submit -91 or a gazillion other negative numbers. Or, while we are into domain stretching pick any real number. It satisfies the criteria.
But this opens up an extended family of interesting numbers – all along the lines of
What is the smallest number that can be expressed as the sum of n mth powers in k ways.
I have explored when n and m are both 2 and k is a (small) natural. It leads to really interesting patterns with obvious pythagorean connections.
Looking at the sum of three squares or the sum of 2 cubes are considerably more difficult. I haven’t made much progress in this direction.
However, the point is, while such searches for interesting properties may appear pointless and quirky at first, they do open up a lot of interesting avenues for mathematical exploration with the result that new and useful mathematical tools may be developed.
Alternately, one might define “interesting” as “can be expressed in fewer bits of information than its direct place-value representation”. In this case, a number like 2^43112609 - 1 is certainly interesting, but the vast majority of numbers, including all of the relatively small ones, are not.
Please explain more! I’m genuinely curious. (And try to kep to about a ninth-grade math level…)
A mathemetician friend of mine knew that he had encountered true love when he was giving directions to the Ren Faire site in Minnesota to a new musician in his consort…
Him: “Just take 13 west and turn south at 169, which is easy to remember, because…”
Her: “169 is 13 squared.”
Not really. Look at Chaitin’s constant (actually since it’s infinite, any lengthy prefix of it). It’s algorithmically random, so cannot be represented by an functional description shorter than itself. But if we knew it’s value, even to a few hundred bits, it would be of tremendous interest.
There is difference between the two paradoxes, though. In the unexpected hanging, you are passing days and therefore eliminating them from consideration. If the criteria for interesting is “smallest non-interesting number”, then you cannot have more than one number qualify by definition.
I suppose a more useful definition that avoid paradox would be “smallest not-otherwise-interesting number”.
Well, we know it to 43 bits for a certain Turing machine (actually, I think the current record is 60, but I wasn’t able to find it on a quick google):
00010 00000 01000 01010 01110 11100 01000 00101 110
The problem is, in order to compute this, you need to check whether all programs (on some Turing machine) shorter than the given length halt, which is exactly the information you get from knowing the number – so through computing omega, you actually don’t learn anything you didn’t know before.
However, if you were just magically told the first couple of hundred bits of some omega number, then yes, you’d be able to solve most outstanding mathematical problems in a flash – but that wouldn’t be any different than an oracle just telling you the answers straightforwardly.
Though if interesting = information content, then I agree with you: the numbers that can be compressed are those that have limited information content, so any incompressible (i.e. random) number would by definition be more interesting. But again, this just shows that ‘interestingness’ is not really a well defined property…
I’m going to my hospital room to die now.
It’s all about St. Stephen, but I’ll give cites later.
Something occured to me, but the coffee hasn’t kicked in yet so my description will probably be worse than even usual.
Small numbers can be interesting because they have properties that only small numbers would generally have. As numbers get bigger they are less likely to be able to compete with the small numbers in the area of “small number type uniqueness”.
However, this opens up new possibilties for lets say typical medium sized unique properties. So they still have some other area in which to be able to compete so to say.
And then on to bigger and bigger numbers.
So, with thinking about it about 5 minutes, while I would not say every number could be described as interesting in some way (depending on how far you stretch the definition), I am inclined to say that the distribution of interesting numbers does not go to zero no matter how large your number gets.
Though, as the number gets bigger its going get hard and harder in a real computational sense to “compute” that interesting fact because you’ll has so much uninteresting stuff to go through as well as having a larger and larger set of other large numbers you have to compare it to to make sure they don’t have the same interesting quantity.
Indeed you can. Allow me to briefly stop work on my Time Cube calculations to explain how:
QED.
A conjecture. The larger the number the more likely it is you can find some interesting characteristic about it. As the size of the number approaches infinity the probability approaches 100 percent.
Easily, just hire Morgan Freeman to narrate it.
But this all depends on the particular encoding you’ve chosen. One could argue – though one could also argue the opposite – that unsigned binary digit representation is the ‘natural’ encoding of a positive integer, and the number of bits needed is a ‘natural’ value for the ‘bits of information in its direct place-value representation’, but there’s certainly no clear obvious natural way of encoding operations like subtraction or raising to a power. There are an infinite number of ways to encode the calculation 2^43112609-1. For instance, we could write it out like we just did and encode the characters as 7-bit ASCII characters (which gives us 84 bits, assuming we don’t need a stop or start signal). Or we could use 7-bit ASCII for operators (“^”, “+”, etc), with a special ASCII character that signals the next X bits are a binary representation of an integer. That gives a different number of bits.
Or we might have some encoding scheme that includes representation of multiplication and subtraction, but not raising to a power. In which case the encoding of Chronos’ Number will be longer than the straight-binary representation.