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.
Challenges:
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!
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.
Challenges:
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!
