Let $v_p(n)$ denotes the largest power of $p$ that divides $n$. Evaluate $v_3(2^{3^n}+1)$. Also Find the generalization of the problem.

Question

Let $v_p(n)$ denotes the largest power of $p$ that divides $n$. Calculate $v_3(2^{{3}^{n}}+1)$. Is there any generalization of it?

My try : Answer is $v_3(2^{3^n}+1)=n+1$. I tried to prove it by induction. For $n=1$ it is trivial. For $n=k$ and $n=k+1$ I have also proved by induction and got the answer as $n+1$. But I got stuck when I was told to find the generalization of the problem. Any help would be appreciated.

Answer 1

Hint

If $\ p\ $ is any prime, what is the largest power of $\ p\ $ that divides $$ (p-1)^{p^n}+1\ ? $$