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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 ?

Peter
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  • This is related to your former question. In the end, I suppose, there remains a brute force part. – Dietrich Burde Apr 05 '17 at 13:29
  • @DietrichBurde Yes, the question is closely related, but I hoped that at least this problem could be solved efficiently. I heard that most techniques used by compter programs rely on conjectures. Would this considerably accelerate the verification ? – Peter Apr 05 '17 at 16:57

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