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# Intuition tells no, but how do you do this?!

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...)

Note by Steven Jim
1 week, 1 day ago

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It 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

- 1 week, 1 day ago

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

Any ideas?

- 1 week ago

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

- 1 week ago

Anything actually come to your mind yet?

- 1 week ago

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.

- 6 days, 17 hours ago

Thanks! I’ll check it out.

- 6 days, 15 hours ago

this is for series i think

- 5 days, 17 hours ago