Prove that any positive rational number $\dfrac{m}{n}$ can be represented as $\dfrac{a^3+b^3}{c^3+d^3}$ where $(a,b,c,d)\in \mathbb{N}^4$.
Can anyone give a full solution please? Unfortunately I have no idea in particular for $m=2$ and $n=1$.
Asked
Active
Viewed 74 times
0
-
Why downvote??? – RFZ Jul 27 '16 at 06:15
-
3Not the downvoter, but after 2000+ rep points surely you can recognize when a question is lacking any context (and also notice that it is phrased like a stereotypical homework request)? – Erick Wong Jul 27 '16 at 06:17
-
7see Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$ – mathlove Jul 27 '16 at 06:38
-
@mathlove, Thanks for link! – RFZ Jul 27 '16 at 06:50
-
The smallest solution for $m/n=2$ comes from $1456/728$. – Gerry Myerson Jul 27 '16 at 07:23
-
@GerryMyerson, How you get these values? Can you explain it more detailed please? – RFZ Jul 27 '16 at 08:12
-
1http://oeis.org/A003325/b003325.txt has a list of numbers that are the sum of two positive cubes. Go down the list, looking at the even ones, checking to see whether half the number is on the list. – Gerry Myerson Jul 27 '16 at 09:03