The $nth$ taxicab number $Ta(n)$ is the smallest number that can be expressed as $n$ unique sums of two cubes. For example, the first non-trivial taxicab number $Ta(2)$ is $1729$, where:
$1729=9^{3}+10^{3}=12^{3}+1^{3}$.
We might agree that taxicab numbers are interesting and worthy of study in their own right. It can also be interesting and useful to be aware of the “practical” applications of math. So, I ask: do taxicab numbers have known applications in the “real world”? (Or do related topics, such as cabtaxi numbers?) Or, are there areas where researchers think there might be applications, but which that idea has not yet been born out?
To try to answer this, I’ve looked at the Wikipedia and OEIS entries on taxicab numbers, and neither address applications. Google searches, (even when excluding the more popular question regarding the unrelated taxicab metric) have also not been fruitful.
