Finding all integer solutions of the following inequality $\bigg| \sqrt[3]{2}-\frac{p}{q} \bigg | <\frac{1}{q^{5/2}}$

I want to find integer solutions of the following inequality by using Mathematica

$$\bigg| \sqrt[3]{2}-\frac{p}{q} \bigg | <\frac{1}{q^{5/2}}$$

Reduce[Abs[Surd[2, 3] - p/q] < 1/q^(5/2), {p, q}, Integers] //ToRadicals // TraditionalForm


but there is a strange result

I also tried FindInstance but there is the only one solution

FindInstance[Abs[Surd[2, 3] - p/q] < 1/q^(5/2), {p, q}, Integers, 2]


There are 3 solutions according the Wolfram Alpha

but...

WolframAlpha["Abs[Surd[2,3]-p/q]<1/q^2.5)", {{"IntegerSolution"}, "Content"}]


{}

It's still not working..

I tried "Open Code" in WolframAlpha but it didn't help me.

what's going on here? :)

• If you don't mind specifying a range of integers to search (which, understandably, you probably do) you could use a region: reg = ImplicitRegion[Abs[Surd[2, 3] - p/q] < 1/q^(5/2), {{p, 1, 5}, {q, 1, 5}}]; and then Reduce[Element[{p, q}, reg], {p, q}, Integers] gives you the desired output. Commented Sep 30, 2017 at 12:46
• A graphical solution is possible with Show[Graphics[{LightGray, Table[Point[{i, j}], {i, 0, 6}, {j, 0, 6}]}, Axes -> True], RegionPlot[Abs[Surd[2, 3] q - p] < 1/q^(3/2), {p, 0, 6}, {q, 0, 6}, Frame -> True], Graphics[{Red, Point[{1, 1}], Point[{2, 1}], Point[{5, 4}]}, Axes -> True]] to make yourself shure this are all solutions of integer pairs. Commented Mar 26, 2020 at 21:56

Union[Flatten[