# Find values of n for which expression is integer

I have an expression for which I want ot find values of n that will make it an integer. Here is the expression:

(n^(3/2) - n)^(1/3)
or
n^(1/2) - n^(1/3)


I want to find all positive values of n for which the above expression will be an Integer. Can I do that with Mathematica?

Thanks.

• You can use "Reduce: "Reduce[{b == (n^(3/2) - n)^(1/3), n > 0, b [Element] Integers}]". This gives an answer using a root object. This is a shorthand for the roots of the enclose polynomial. If you want to transform it to radicals, you give to output to: "ToRadicals" – Daniel Huber Nov 8 '20 at 20:00

Using InverseFunction

f := Evaluate@InverseFunction[(#^(3/2) - #)^(1/3) &]

a = N[f /@ Range[3]]


{2.1479, 5.73535, 11.3776}

The above values will produce integers, e.g.

(#^(3/2) - #)^(1/3) & /@ a


{1., 2., 3.}

• Instead of f := Evaluate@InverseFunction[... you could simply write f = InverseFunction[.... – Roman Nov 8 '20 at 21:05
• @Roman Ah, yes. With Set rather than SetDelayed. – Chris Degnen Nov 8 '20 at 21:39