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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 \(T^{*}(4) = 6963472309248\).

So I ask:

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

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
1 year, 8 months ago

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How did you derive \(T^{*}(3)\) and \(T^{*}(4)\)? Julian Poon · 1 year, 8 months ago

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@Julian Poon 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. Jake Lai · 1 year, 8 months ago

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@Julian Poon nvm... heres the article Julian Poon · 1 year, 8 months ago

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