For $a,b,c,m,n$ are positive integers, prove that: We can write a positive rational number $q$ in form : $ q=\dfrac{a^3+b^3}{m^2+n^2}$.

This problem is true when we change $m^2+n^2=m^3+n^3$.

  • $\begingroup$ I'm not sure I understand the question. $\frac{a^3+b^3}{m^2+n^2}$ will always be a rational number when $a$, $b$, $m$ and $n$ are integers $\endgroup$ – tpb261 Jun 24 '14 at 10:58
  • $\begingroup$ @tpb261 If q is a positive rational number it can be written as $\frac{a^3+b^3}{m^2+n^2}$ where $a,b,m,n$ are positive integers $\endgroup$ – kingW3 Jun 24 '14 at 11:00
  • $\begingroup$ The formula will be cumbersome. For more simple equation can be viewed there. math.stackexchange.com/questions/369846/… $\endgroup$ – individ Jun 24 '14 at 11:48
  • $\begingroup$ We can change $m^2+n^2=m^2+n^3$ $\endgroup$ – user41499 Jun 24 '14 at 14:03
  • $\begingroup$ And what's the point? You still need to solve this equation. And more. $\endgroup$ – individ Jun 24 '14 at 15:15

There is an obvious way to do this, and in fact you can take $a = b$, $m = n$. If $q = c/d$, simply take $a = b = c d$, and $m = n = c d^2$.


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