Let $m,n$ be positive integers, such that $m^2\neq n^3$. I conjecture the inequality
$$|m^2-n^3|>\dfrac{1}{5}\sqrt[6]{m^2+n^3}.$$
I've tried a lot of numbers, and they all seem to work, but how do I prove it?



On the conjecture : $|m^2-n^3|>\frac{1}{5}\sqrt[6]{m^2+n^3}$
As suggested by **Joe Silverman**, there are counterexamples in my paper



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> Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, *Lecture Notes in Computer Science* **1838** (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63. arXiv: math.NT/0005139 (<https://arxiv.org/abs/math/0005139>)
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The best one there, which I think still holds the record for the largest ratio
$n^{1/2} / |m^2-n^3|$ known, is
$$
\begin{array}{rcl}
m & \!\! = \!\! & 447884928428402042307918, \cr
n & \!\! = \!\! & 5853886516781223,
\end{array}
$$
with
$$
|m^2-n^3| < \frac{1}{52.3}\sqrt[6]{m^2+n^3}.
$$


