Rationals. (1 Viewer)

[Fus Ro Dah]

 

[Sy123]

Are there non-integer rational solutions?

 

[seanieg89]

There are no non-integer rational solutions by the rational root theorem. In fact I claim that the only rational solutions are: (0,0), (0,1), (1,0), (1,1).

Outline of proof:

-Establish the result for non-negative integers using that x^3 >= x^2 for non-negative integers with equality iff x=0 or 1.

-Explain why pairs of negative integers cannot be solutions.

-Let f(t)=t^3-t^2. It remains to show there are no solutions to f(x)=-f(y) with x a non-neg int and y a negative int.

-We may assume x is positive in fact, as the case x=0 is trivial and has no solutions.

-Observe that f is increasing on [1,inf)

-Observe that f(-y) < -f(y) < f(1-y).

This sandwich is enough to complete the proof using the previous observation.

 

[Carrotsticks]

 

[seanieg89]

My bad, apologies for the rushed and incorrect work.