The equation $y^2=x\pm \ell $

Question

For what odd primes $\ell$ does the equations $y^2=x\pm \ell$ have a finite set of solutions over the integers. Here, assume $y$ is even and $x$ is a prime number. I am not sure if this is really hard or no but the most obvious thing we see is that $x\equiv \mp \ell [4]$. Otherwise, I don't know how to proceed. Thank you for any help / hint !

Answer 1

The answer is unknown even in the case $y^2=x-1$ (Landau problem). It is unknown for all non-obvious cases also.An obvious case is an equation of the form $y^2= x+r^2$. You can read about the more general Bunyakovsky conjecture here.