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?

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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?

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

- Bigtone53
**Posts:**413**Joined:**19 September 2005

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.

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.

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.

- Bigtone53
**Posts:**413**Joined:**19 September 2005

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".

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

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

- Glyn
**Posts:**357**Joined:**26 April 2007

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.

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.

- udosuk
**Posts:**2698**Joined:**17 July 2005

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.

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.

- Bigtone53
**Posts:**413**Joined:**19 September 2005

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