# Is $577^3$ the smallest cube that is expressible as the sum of $3$ positive cubes in $6$ different ways?

A Taxi Cab Number is a number $\text{Ta}(n)$ that can be written as the sum of two cubes in $n$ different ways. More formally, they are known as Hardy-Ramanujan Numbers or, as Ramanujan had called them, Magic Numbers.

\begin{align} 577^3 &= 356^3 + 385^3 + 448^3 \\ &= 1^3 + 426^3 + 486^3 \\ &= 172^3 + 318^3 + 537^3 \\ &= 41^3 + 244^3 + 562^3 \\ &= 153^3 + 174^3 + 568^3 \\ &= 90^3 + 201^3 + 568^3 \end{align} Is this the smallest cube that is expressible as the sum of $3$ positive cubes in $6$ different ways? In symbols, does 577^3 = \text{taxicab}(3, 3, 6)\,?\tag*{\bigg(\begin{align} \verb|S|&\verb|uch that the LHS| \\ &\verb|must be a cube.|\end{align}\bigg)} I have been trying to find numbers similar to Taxi Cab Numbers. The best example of a taxicab number is $1729$ because it is the smallest number that can be expressed as the sum of two cubes in two different ways, i.e. $$1729 = 1^3 + 12^3 = 9^3 + 10^3.$$ I developed a more general case of finding numbers $a_n^{ \ \ 2n - 1}$ that can be written as the sum of $(n+1)$ cubes in $2(2n-1)$ different ways, trying to find a potential pattern in the values of $a_n$ for which $a_n \in\mathbb{Z^+}$.

Given that $n = 1$, we obtain that $a_1 = 1729$. Given $n = 2$, we apparently obtain $a_2 = 577$. If this is true, the only pattern I can find is that $$a_n^{\ \ 2n - 1} - 1 = \left\{\sum_{k=1}^n b_k^{\ \ 3} : b_k\in\mathbb{Z^+}\right\}.$$ Also, could somebody find the value of $a_3$ and $a_4$ and if they want, $a_5$ (though I think the smallest number that is the sum of $9$ cubes in $18$ different ways might be pretty large).

P.S. I was not able to find any other appropriate tags apart from > (number-theory) <

• Is this the smallest cube that is expressible as the sum of $3$ cubes in $6$ different ways? In symbols, does $$577^3 = \text{taxicab}(3, 3, 6)\,?$$ This is not exactly the same question: taxicab(3,3,6) may not be a cube itself. – Arnaud Mortier Feb 25 '18 at 11:09
• @ArnaudMortier true that. Perhaps I will make it clearer. Thanks for pointing that out :) – Mr Pie Feb 25 '18 at 11:21
• $577^3$ is certainly not the smallest cube that can be expressed as a sum of three cubes in six different ways, as $1^3$ can be expressed as a sum of three cubes in infinitely many different ways – see, e.g., math.stackexchange.com/questions/32559/… But perhaps you meant a sum of three positive cubes. If so, please edit your question to reflect that intention. – Gerry Myerson Feb 25 '18 at 11:57
• @GerryMyerson yes I did, because I knew of the one with $1^3$ so thank you for reminding me :) – Mr Pie Feb 25 '18 at 12:42
• @GerryMyerson I commented here $\longrightarrow$ mathoverflow.net/questions/138886/… – Mr Pie Feb 25 '18 at 12:51

\begin{align}216^3 &=24^3+ 144^3+ 192^3\\ &=26^3 +102^3+208^3\\ &=30^3 +164^3+ 178^3\\ &=48^3 +76^3+ 212^3\\ &= 102^3 +117^3 +195^3\\ &=108^3 +144^3+ 180^3\end{align}
• Thank you very much! But just out of curiosity, if $a_2^{\ \ 3}$ had to strictly be a cube number, and not like $216^3$ because that is equal to $6^9$ as well, then would $a_2 = 577$? I cannot vote now because I have reached my daily limit, so I have to wait a couple of hours. – Mr Pie Feb 25 '18 at 12:45