# A weird question

I'm sure almost all of you are familiar with Ramanujan's number, 1729. What about 87539319, or 6963472309248?

Define the $n$th strict taxicab number $T^{*}(n)$ as the smallest number which can be expressed as the sum of two positive integral cubes in exactly $n$ ways. Then, it turns out that $T^{*}(3) = 87539319$ and $T^{*}(4) = 6963472309248$.

$\text{Is the sequence } \lbrace T^{*}(n) \rbrace_{n=1} \text{ strictly monotonically increasing?}$

Obviously (I think) this is not a question we can currently answer. I don't even know if the sequence is bounded above. Still, I think it's a pretty fun question to ask as a sort of time capsule, to look at how far the tools of mathematics will sharpen in the decades to come.

Note by Jake Lai
4 years, 8 months ago

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How did you derive $T^{*}(3)$ and $T^{*}(4)$?

- 4 years, 8 months ago

nvm... heres the article

- 4 years, 8 months ago

Actually, the regular (nonstrict) taxicab numbers $T(n)$ is the most common definition. For small $n$ it corresponds to $T^{*}(n)$. I was thinking about whether or not it was possible if $T^{*}(n) > T^{*}(n+1)$ for large $n$ despite the latter being expressible in one more way than the other.

- 4 years, 8 months ago