I haven't find a concrete supportive example nor a proof of non-existence of such a representation yet.

But there are still some clues :

……………………………………………………………………………………………………………

First, realize \(2018^{2}\equiv 4 (mod9)\),

and \(\forall x\in Z,x^{3}\equiv 0,\pm 1(mod9)\) .

So **if** \(2018^{2}\) can be represented as \(a^{3}+b^{3}+c^{3}+d^{3}\) for some \(a,b,c,d\in Z\),

we certainly get \(a^{3}\equiv b^{3}\equiv c^{3}\equiv d^{3}\equiv 1(mod9)\).

And further, \(a,b,c,d\in \{9k+1|k\in Z\}\cup\{9k+4|k\in Z\}\cup\{9k+7|k\in Z\}\). ………………………………………………………………………………………………………………

**_**' Can,or cannot,this is a question! '

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TopNewestThis is still an open question. See (2) and (3) here.

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Thank you,sir Pi Han Goh. My note was inspired by a problem in 2002 IMO proposal

━ Find the smallest positive integer n satisfying : the diophantine equation \(x_{1}^{3}+x_{2}^{3}+...+x_{n}^{3}=2002^{2002}\) have integer solution ,

whose answer is 4.

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Ah good to know.

Originally, I spent over an hours trying to come up with a proof for your question that no solution exists via cubic residues but I failed badly so I decided to look it up.

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I'm sorry but you might fail to look it up.

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