Adventures with 1729

Talking of cubes as I was in my last post, even when people are thinking about the same thing, they are probably thinking in different ways. A famous English mathematician called Hardy once visited a very clever Indian mathematician called Ramanujan in hospital.

He probably wanted to know if Ramanujan had discovered a curious set of numbers called Carmichael numbers, so Hardy said casually that he had travelled there in a taxi with a boring number, 1729.

Ramanujan sat up in bed. "Oh no, Hardy, on the contrary, that is a very interesting number. It is the first number which is the sum of two cubes in two different ways!"

When Hardy thought it through, he realised that 1729 was the sum of 9 cubed and 10 cubed (729 + 1000), or of 12 cubed and 1 cubed (1728 + 1). When I came across this story, I noticed something much less interesting: 1729 is the product of three prime numbers which are in mathematical progression (7 x 13 x 19) -- I guess I am just a bit boring where numbers go.

One day, I had an idea for a problem: using just the numbers 1, 7, 2 and 9, in that order, and using any mathematical notation, I set out to see how many different numbers I could get.  I began with trial and error, a good old-fashioned method that has always served humanity well.  Bang the rocks together, notice what makes useful flakes, and bang them again, that sort of thing.

That got me a few answers, like 1 = 1^729, or 1 = 1 + 7 + 2 - 9 or 6 = 1 + (7x2) -9 but then I began running out of new numbers.  Then I switched to the sort of clever trick that is used when pharmacists take drugs that work, and fiddle with them, changing the bits systematically.

You can find some hints by poking around in this area. DO look around, or you will miss some important stuff.

 

Then we could turn to the problem that I will describe in my next post, and look at the tabulation method that I use there. . .