Can $2018^{2}$ be represented as $a^{3}+b^{3}+c^{3}+d^{3}$ ? Here $a,b,c,d\in Z$

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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Comments

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

Pi Han Goh - 3 years, 4 months ago

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.

Haosen Chen - 3 years, 4 months ago

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.

Pi Han Goh - 3 years, 3 months ago

@Pi Han Goh – Oh,I'm sorry,sir. I said I was inspired because $2002^{2002}=(2002^{667})^{3}\cdot (1000^{3}+1000^{3}+1^{3}+1^{3})$ and then I wonder if $2018^{2}$ can be done like $2002^{1}$ ,since $2018\equiv 2(mod3)$ . So …

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

Haosen Chen - 3 years, 3 months ago

@Haosen Chen – No, I totally got it. "I looked it up" = I give up because I don't know how to solve it, so I tried the internet for the answers.

Pi Han Goh - 3 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}`	$\boxed{123}$

Can 201822018^{2}20182 be represented as a3+b3+c3+d3a^{3}+b^{3}+c^{3}+d^{3}a3+b3+c3+d3? Herea,b,c,d∈Za,b,c,d\in Za,b,c,d∈Z

Comments

Can $2018^{2}$ be represented as $a^{3}+b^{3}+c^{3}+d^{3}$ ? Here $a,b,c,d\in Z$