The number 1729 is very dear to the hearts of mathematicians.
The story goes back to a 1919 conversation between the famous British mathematician G. H. Hardy and the Indian genius mathematician Srinivasa Ramanujan. Ramanujan asked the taxi number that Hardy had ridden in on the way. Hardy replied that it was number 1729 and mentioned that the number “seemed to be rather a dull one”.
“No”, Ramanujan replied, “it is a very interesting number; it is the smallest number expressible as the sum of two [positive] cubes in two different ways.”
And it’s true. The number 1729 can be expressed as 13 + 123 and also as 93 + 103 and is the smallest number for which that is true.
In the world of recreational mathematics — yes, there is such a thing — such numbers are now known as Taxi Cab Numbers. They even have their own web site.
The next number in the sequence is 4,104 = 23 + 163 = 9 3 + 15 3, then 13,832 = 23 + 243 = 183 + 203, then 20,683 = 103 + 273 = 193 + 243 and so on.
This gives rise to a variety interesting math problems. For example, can you write a computer program that calculates such numbers? Sure. In fact, here are 25 of them.
The series is infinite. In other words, given enough computing power, you will always be able to find a next-higher number. Always.
And this why stop at two different ways? For example, what’s the smallest number that can be expressed as the sum of two cubes in three different ways? That number is 87,539,319 which is 2283 + 4233 and 1673 + 4363 AND 2553 + 4143
How about in four ways? It’s 6,963,472,309,248, which is 13,3223 + 16,6303 and 10,2003 + 18,0723 and 5,4363 + 18,9483 and 2,4213 + 19,0833 .
You can imagine, they get crazy-big after that.
Oh, why stop with just adding two numbers together? What about adding three numbers? Why not include negative numbers? And exponents other than cubes?
In other words, the sum of A numbers raised to the power of B, C different ways.
Yep. There are an infinite number of all these variations. I didn’t even get into the possibility of subtracting numbers, not just adding them.
That’s the cool thing about math. Almost everything in math is infinite. No matter what cool thing you find, somebody with enough imagination — and perhaps enough computer power — will be able to figure out the next thing one bigger.
Want a billion digits of pi? Okay. Heck, how about five billion?
How about ten million digits of the square root of two?
I could do this all day.
Numbers extend forever. And since numbers are really just a construct of our mind, you could argue that the mind could extend forever.
And only you can decide if that’s a comforting thought ... or a scary one.