A weird question

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

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

So I ask:

Is the sequence {T(n)}n=1 strictly monotonically increasing? \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
6 years, 5 months ago

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

Julian Poon - 6 years, 5 months ago

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nvm... heres the article

Julian Poon - 6 years, 5 months ago

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

Jake Lai - 6 years, 5 months ago

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