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?