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I think most people know these numbers. Find $x,\ y,\ z,\ w$ such that $x^3 + y^3 = z^3 + w^3$ and $x,\ y,\ z,\ w$ are not equal to each other.

The first is $1729$.

I'm trying to figure out if there's a formula/expression to show that the $n^{\text{th}}$ taxicab number is less than some number, but the $(n+1)^{\text{th}}$ taxicab number is greater than it. Any ideas?

For a while, I was thinking that it was $1729\cdot 2^{n-1}$, which works for the first $20$, but not aftewards...

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    I suspect the population that knows the phrase "taxicab numbers" in this sense is a minority of the world population.2012-03-22

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