The New Sudoku Players' Forum

[Bigtone53]

While not wishing to interfere with the other Brainteasers chain, here is something slightly more mathematical. I do not claim it to be complicated.

20683 is the fourth member of an increasing series of numbers. What is the common vehicle here?

[udosuk]

I'm guessing the driver of the vehicle is named Travis or Robert?

[Bigtone53]

Sure is. That was quick! Well done

[wintder]

For us dummies, what?

[Bigtone53]

20683 is the fourth taxi number, so named after a famous conversation between mathematicians G H Hardy and Ramanujan described below
by C P Snow.

“Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”

Taxi numbers or Ramanujan numbers are therefore positive integers that can be expressed as the sum of two or more different cubes in two or more different ways. Sticking to two cubes and two ways, the first 5 numbers are 1729, 4104, 13832, 20683, 32832.

Travis Bickle was the lead in the film Taxi Driver and played by Robert De Nero.

[Glyn]

Using the definition of the Taxicab number Ta(n) from Wikipedia as: the smallest number expressable as the sum of 2 positive cubes in n distinct ways.

The series now reads 2,1729,87539319,6963472309248,48988659276962496,24153319581254312065344,

The sparsity of Taxicab numbers is a result of the famous "Livingstone Congesture".

[Bigtone53]

Ahh, but I did not say taxicab numbers. These are a special subset of taxi numbers, where as you say, number of methods is 2 but number of cubes varies. I am not sure that 2 counts in any definition.

[udosuk]

Here are 2 sources:

http://mathworld.wolfram.com/TaxicabNumber.html

http://en.wikipedia.org/wiki/Taxicab_number

You can find Bigtone's sequence in the 1st source, and Glyn's 2 in the 2nd source.

Note "Taxi numbers" is almost impossible to google for the mathematical meaning as it has too prominent a daily life usage.

[Glyn]

I was looking for an excuse to mention the admirer of 'Newt'on.

[Bigtone53]

Glyn, Your location as London explains everything. 30 years of commuting there also tells me a thing or two!

[Bigtone53]

Back to the maths for those who like such a thing.

Points on a circle. 1,2,4,8,

next ... and the one after

[udosuk]

This might not be what you're after, but I find it also fascinating:

1,2,4,8,16,31,57,99,163,256,...

[Bigtone53]

Perhaps I should save time by skipping the question bit and just posting the answer

Yup, this is Moser's Problem . I find it fascinating that having gone 'off-track' at 31, it is back with a power of 2 at 256.

[Glyn]

Perhaps you should call it Moser's Poser.

I wonder if there any more powers of 2, if so they appear to be a long way further in.