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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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Related: math.stackexchange.com/questions/2815/… –  Aryabhata Mar 21 '12 at 23:56
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What's a "formula" to you? There's clearly an algorithm for it, so if you allow Diophantine equations and some appropriate way to insert quantification over the natural numbers in your formulas, then a (somewhat monstrous) formula can be derived using Hilbert's-10th-problem methods. –  Henning Makholm Mar 21 '12 at 23:57
    
Basically, we're working on a problem where we need to find the Nth taxicab number in less than 1 minute using java. We have the code working except for a bit of efficiency. We want to skip over any case where x^3 + y^3 is going to be bigger than the nth taxicab number. –  Zack Stewart Mar 22 '12 at 0:00
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You might look at the references in oeis.org/A001235 –  Ross Millikan Mar 22 '12 at 0:03
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I suspect the population that knows the phrase "taxicab numbers" in this sense is a minority of the world population. –  Ross Millikan Mar 22 '12 at 4:35
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