# We can write a positive rational number $q$ in form : $q=\dfrac{a^3+b^3}{m^2+n^2}$.

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 $m^2+n^2=m^3+n^3$.

• 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 – tpb261 Jun 24 '14 at 10:58
• @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 – kingW3 Jun 24 '14 at 11:00
• The formula will be cumbersome. For more simple equation can be viewed there. math.stackexchange.com/questions/369846/… – individ Jun 24 '14 at 11:48
• We can change $m^2+n^2=m^2+n^3$ – user41499 Jun 24 '14 at 14:03
• And what's the point? You still need to solve this equation. And more. – individ Jun 24 '14 at 15:15

## 1 Answer

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 $m = n = c d^2$.