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.

#NumberTheory #Sequences #Ramanujan #Hardy-RamanujanNumber #TaxicabNumbers

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

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

Julian Poon - 6 years, 3 months ago

nvm... heres the article

Julian Poon - 6 years, 3 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.

Jake Lai - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$