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# 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
2 years, 5 months ago

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How did you derive $$T^{*}(3)$$ and $$T^{*}(4)$$? · 2 years, 5 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. · 2 years, 5 months ago