Show that all positive rational numbers can be written in the form
$$\frac{a^3+b^3}{c^3+d^3}$$
where $a,b,c,d$ are positive integers.
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$\begingroup$ The answer is here: math.stackexchange.com/questions/1185535/… $\endgroup$– timon92Commented Feb 13, 2017 at 15:14
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$\begingroup$ See here: artofproblemsolving.com/community/c6h19763p131812 $\endgroup$– XamCommented Feb 13, 2017 at 18:18
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$\begingroup$ Thanks all of your comments, they really helps! $\endgroup$– MathelogicianCommented Feb 13, 2017 at 18:55
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1 Answer
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Xam commented on 2017 Feb 13 that an answer was given here by zabelman on 2005 Dec 17. Also on 2017 Feb 13, timon92 mentioned the following answer found in a post by mathlove on 2015 Mar 11:
For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{c^\color{red}{3}+d^\color{red}{3}}.$$
For $r=p/q$ where $p,q$ are positive integers, we can take $$(a,b,c,d)=(3ps^3t+9qt^4,\ 3ps^3t-9qt^4,\ 9qst^3+ps^4,\ 9qst^3-ps^4)$$ where $s,t$ are positive integers such that $3\lt r\cdot(s/t)^3\lt 9$.
For $r=2014/89$, for example, since we have $(2014/89)\cdot(2/3)^3\approx 6.7$, taking $(p,q,s,t)=(2014,89,2,3)$ gives us $$\frac{2014}{89}=\frac{209889^3+80127^3}{75478^3+11030^3}.$$