I'm sorry to ask this question probably it is not suitable here. However I read some papers in number theory related to the solution of $x^3+y^3+z^3=n$ for some known solved problem about representation of numbers as sum of three cubic like $42$. But all of those representations never used Wolfram Alpha as a means of solving. I'm interested in that problem and at the same time I do not want to waste my time to search about triplet $(x,y,z)$ for which $x^3+y^3+z^3=390$ using Wolfram Alpha if it is not a suitable way for searching.
Now my question here is: Can someone find the solution of $x^3+y^3+z^3=390$ using just Wolfram Alpha? Or it is not suitable means for that? Or we should wait solution from supercomputer?
How do you know solutions exist? This should always be the first step. You will never disprove that there are no solutions (in this case) by trying 'lots' of numbers. [I hear the words of John Conway talking about 'proving Fermat's Last Theorem by inspection' when asked by a reporter echo in my ears.] Even if you know they exist, unless you have bounds on the solutions or another some other better method, they are not likely to be 'easily' (if at all) found by computational 'guess 'n check'.
Even if this is the way to go, WolframAlpha will NEVER be the way to do this. It will time you out after less than 30 seconds of computation. So you better hope that whatever you are trying is done by then. I think you mean to ask if you would use Mathematica to check? In which case, I think it depends on the researcher. Most that I know will use a more fundamental programming language like C++ or Python and then parallel process because you want the speed. Otherwise, everything I know being done in Computational Arithmetic Geometry is done in Magma or Sage for the nice balance of speed and built in functions.
EDIT. I should have been perhaps more clear. Depending on the problem, it may not even matter the computer, and that 'guess 'n check' will never be the approach (this is almost always the case anyway). Take an imaginary computer $10,000$ times more powerful than the world's current greatest supercomputer - or even one the size of the sun - it makes no difference, some examples/counterexamples are so computationally large or expensive that you will never find them by 'guess 'n check' or blindly computing away. Take for example Tree$(3)$ or Graham's Number, or a better example from the same general area of Math you are asking about, take Skewes' Number in Number Theory. To learn about this, why not check out this nice brief summary.