Despite of the negative result of Hilbert's tenth problem, diophantine equations of the form $$f(x)=y^n$$ can often be fully solved. Here, $f(x)$ is a polynomial with integer coefficients and $n>1$ an integer.
Is there a systematic way to determine which techniques can be applied ?
I am aware of some techniques like Elliptic curves or hyper-elliptic curves. But I never know which technique can be applied in a concrete case. An example is $$x^2-x+1=y^n$$ which I would like to solve over the positive integers , where $n\ge 2$ is a parameter. This would solve the special case of my question when $$\varphi(n)^2+n$$ is a perfect power. For the primes the expression is $\ p^2-p+1\ $.
$\ x=1\ $ and $\ x=19\ $ are the only solutions upto $10^8$ for every $n$. Any ideas for this special cases ? Any general advices for other functions $f(x)$ ?