# Number Theory Question from Swedish Maths Olympiad: Find all integers $n\geq 8$ such that $n^{\frac{1}{n-7}}$ is also an integer.

I encountered this video about number theory question from Swedish Maths Olympiad:

Find all integers $$n\geq 8$$ such that $$n^{\frac{1}{n-7}}$$ is also an integer.

The video shows step by step solution. However, I think the question can be solved much easier as the following:

We substitute $$x=n-7$$ and $$n=a^{x}$$ to obtain $$a^{x}=x+7$$. Then quite obviously we find that for $$x\geq4$$ we have $$a<2$$ and we only need to evaluate $$x=1,2,3$$.

Then we find that the only pairs $$(a,x)$$ are $$(8,1)$$ and $$(3,2)$$ i.e. the only solutions are $$n=8,9$$.

Here is the link of the video: Swedish Mathematics Olympiad

• +1 looks flawless to me. Commented Sep 30, 2020 at 22:08
• 9/(9 - 7) is not an integer. Commented Oct 1, 2020 at 1:29
• @WilliamElliot do you mean $9^{\frac{1}{9-7}}=3$? Commented Oct 1, 2020 at 1:32
• You forgot to formulate explicitely your question. Commented Jul 23, 2023 at 13:31