# 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. – aardvark2012 Sep 30 '17 at 12:46

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