For an integer \(a>1\), can \(\sum _{ n=1 }^{ k }{ \sqrt [ a ]{ n } } \) ever be an integer for any integer \(k>1\)?

For \(a=2\), the answer is most likely no. However, I don't have one single clue of how to prove this with \(a=2\), let alone the generalization!

Is there a definitive answer for all integer \(a\)? If so, how?

(It might have something to do with Calculus here, it looks like some kind of series...)

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## Comments

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TopNewestIt doesn't make sense to me, do you mean the n to be an x? Otherwise, you are just multiplying the root by the value of k

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Thanks for your info. I didn't have much time when I was typing down.

Any ideas?

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I agree that it is probably no, but not sure how to prove it

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While doing some research about this, I came across this blog post from almost a decade ago, which applies to your problem for \(a = 2.\) As for higher values of \(a,\) I haven't found anything yet, but perhaps the methods in the blog post can help.

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Thanks! I’ll check it out.

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this is for series i think

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