25 July 2013

Interesting numbers

There's a sort-of theorem in mathematics that all numbers are interesting.*  But I'm thinking first of 1729, which is the subject of a story about how it is an interesting number.  The story is that Ramanujan, a brilliant, self-taught mathematician, was in the hospital (he died at only 32).  G. H. Hardy visited him and commented that his taxi cab was number 1729, which wasn't a very interesting number.  Ramanujan replied that it was indeed interesting -- it was the smallest number that was the sum of two cubes, in two different ways.  That is, 10^3 + 9^3 = 12^3 + 1^3 = 1729.  This number appeared also on a cab in an episode of the Simpsons.

There's a different bit of playing with numbers, one of the longest-unproved theorems in mathematical history.  That is Fermat's Last Theorem.  (Itself misnamed, as he never showed a proof for it, and he worked for years after stating it.)  That is, if we use only integers (1,2,3,...), the equation x^n + y^n = z^n has no solutions for n > 2.  He said this in 1637, and it wasn't proven until 1995.

Let's look specifically at n = 3.  I can rewrite Ramanujan's example as:
x^3 + y^3 = z^3 + 1  (where he had x,y,z = 10, 9, 12)
Fermat's equation is:
x^3 + y^3 = z^3   -- and this has no solutions for integers.

That's interesting -- such a small change, and we go from having no solutions, no matter how large we make x,y,z, to having ... how many?  Well, that's a question.  My version here is a more specialized version of so-called 'taxicab numbers' (named in honor of the above story).  You can see some more about them at Durango Bill's.  But I like mine better because of the connection to Fermat's Last Theorem. 

It seems common that answers in mathematics are either 0, 1, or infinity.  Fermat's equation (for n > 2) has 0 solutions.  We already have 1 for my 'Fermat-Ramanujan' equation.  If there's another, not that this is a proof, probably there are an infinity.  So I set my computer to some brute-force searching, and indeed there are more.  Not many.  It found 92 for z going from 12 (the smallest that has a solution) to 2,000,000 (which was pushing the limit of the computer; z^3 at that point is 8,000,000,000,000,000,000).  That suggests that there are an infinity of solutions to the Fermat-Ramanujan equation (a name I just invented, as far as I know), by that rule of thumb.

Can you find some? 
Can you do it by a more elegant method than having a computer pound away?
Can you prove that there _are_ (or are _not_) an infinity of solutions?

I can say that from my search, the solutions are getting pretty sparse as the numbers get larger.  Maybe the step to the next solution goes to infinity?  i.e., there isn't a 'next' one at some point.  Brute force computing can't answer that.  Need some real thought for the job.

* The 'proof' that all numbers are interesting:
0 is interesting because it is the basis for place value mathematics, or because it's the number you can add to any number and get back the original number, or ...
1 is interesting because you can multiply any number by it and get back the original number.
2 is interesting because it is the first prime number (can only be formed by multiplying itself by 1), is the only even prime number, is the first even number, ...
3 is the first odd prime number, ...
4 is the first perfect square number
and so on.  It's quite a while before it starts to be hard for a numbers person to find a number interesting.

Now suppose we keep going up the list, and finally find a number that isn't interesting (for being prime, a perfect square, a perfect cube, ..., a perfect number, a palindromic number, a taxicab number, ...).  Well, then, _that_ would be interesting!  There are so many ways for a number to be interesting, and this number avoids them all.  That's pretty interesting itself!


Kevin O'Neill said...

If you find numbers interesting, then you should visit The Online Encyclopedia of Integer Sequences.

Type in 1729 and you'll get lots of ideas :)

Robert Grumbine said...


Not just interesting, but 525 different ways that the site finds it interesting.

72, on the other hand, shows up 15,767 different times.

Thanks for the link.

Kevin O'Neill said...

If you haven't looked at OEIS A050792 yet, it has a useful comment:

"One of the simplest cubic Diophantine equations is known to have an infinite number of solutions (Lehmer, 1956; Payne and Vaserstein, 1991). Any number of solutions to the equation x^3 + y^3 + z^3 = 1 can be produced through the use of the algebraic identity (9t^3+1)^3 + (9t^4)^3 + (-9t^4-3t)^3 = 1 by substituting in values of t. ...
"Although these are certainly solutions, the identity generates only one family of solutions. Other solutions such as (94, 64, -103), (235, 135, -249), (438, 334, -495), ... can be found. What is not known is if it is possible to parameterize all solutions for this equation. Put another way, are there an infinite number of families of solutions? Probable yes, but that too remains to be shown." Herkommer