Suppose, we know an integer solution $(y,x)$ of a Mordell-curve $$y^2=x^3+k$$
For example $$25124268633183975113^2=8578189162349^3-251669431780$$
So for $$k=-251 669 431 780$$ , the pair $$(25 124 268 633 183 975 113, 8 578 189 162 349)$$
is a solution.
Can we check efficiently (without brute force) whether there is an integer solution with a smaller $x$ for the given $k$ ? In other words, can we check efficiently whether the solution is the smallest ?