Diophantine eqn..

y=√(x^3 -2)
therefore
domain of y =[2^(1/3),∞)
now
dy/dx=(1/(2√(x^3)-2))*3x^2
which is always positive
therefore always increasing
so the graph of this equation will never touch x axis
so there can be no integral solution

Interesting... the same mistake I did a long time back:

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=220349

The point which you should realize is that we do not require y to be zero but rather we require y to be an integer.

The only integral solutions for (x,y) are (3, ±5).
This is a famous problem due to Fermat and I know of the solution due to Euler. However, the proof is exceedingly difficult. So I won't post it until entirely indispensable.

P.S. This is an example of the so called Mordell's equation: y² = x³ + k which was explored around 1600's. However, a solution for few values of k was not discovered untill 1900's when the theory of algebraic curves were developed. The general solution for an arbitrary k is not yet known.

so we can just fire an arrow in the dark....and hope that it hits the target.........nice...