Taxicab Numbers
I got a little distracted halfway through my number theory homework so I decided to write a post. At least the topic is pertinent to number theory, right?
I first came across taxicab numbers a few years ago while jumping between Wikipedia math articles. Last summer, I was watching the play “Proof” (which was amazing by the way, if you’re ever in New York while its playing then I recommend you watch it) with a friend of mine and the main character mentioned that “1729” is a special number. As soon as I heard that, I felt a memory trying to make its way out from the back of my brain. After a few seconds I finally remembered where I’d seen it and I started excitedly whispering to my friend, frantically trying to explain why the number was special (she still makes fun of how workedup I got).
For those that are unfamiliar, 1729 is known as the HardyRamanujan number after this anecdote:
“I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”  G.H. Hardy Source
How does this related to taxicab numbers? In mathematics, the n^{th} taxicab number is the smallest number that can be written as the sum of two positive cube numbers in “n” unique ways.
Unfortunately I must get back to this problem set, so I just quickly threw together a program in R to test whether a number can be expressed as the sum of two positive cube numbers in 2 different ways. The script doesn’t test whether such a number is the smallest so it is not a true taxicab check in that regard. Perhaps I’ll update this later to be a true taxicab check or even write a program to generate taxicab numbers (this seems like a difficult and fun problem).
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Some examples (the function runs very fast):
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