The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$. 
The number of  solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$.

I was trying to show that this equation actually has infinitely many solutions. I felt that Hensel's lemma will be helpful here, but I cannot figure out how to apply it.
Any comments are appreciated! Thanks in advance!
 A: Edit Correct a typo and make things more precise. 
We want to show that for any $N$ big enough and any $a\in \mathbb Z_p$, there exists a solution with $y=p^Na$ and $x\equiv 1 \mod p$. 


*

*If $n$ is prime to $p$, then for any $b\in\mathbb Z_p$, the equation $(1+pw)^n=1+pb$ has a solution with $w\in \mathbb Z_p$. Or, if you prefer,
$1+pb$ has a $n$th root in $\mathbb Z_p$, congruent to $1$ mod $p$. 

*If $n=p$, then for any $b\in\mathbb Z_p$, the equation $(1+pw)^n=1+p^2b$ has a solution with $w\in \mathbb Z_p$. 
Both statements are proved by developping the left-hand side, simplifying by $1$ and dividing by $p$ (resp. $p^2$) to get an equation 
$$ F(w)=0$$
with $F(w)\in \mathbb Z_p[w]$ (not monic) and $F(w)\equiv w+b \mod p$· Then apply Hensel Lemma which states that there is a solution. Moreover, inspecting the form of $F(w)$, we see that any solution satisfies $v_p(w)=v_p(b)$.
For a general $n$, write $n=mp^r$ with $m$ prime to $p$. Using repeatly (1) and (2) above, we see that for any fixed $a\in\mathbb Z_p$, the equation $$(1+pw)^n=1+p^{2r+1}a$$ 
has a solution with $w\in\mathbb Z_p$. 
Now back to your equation. Fix any $a\in \mathbb Z_p$ and any $N\ge 2$. We are looking for solutions of the form $y=p^Na$ and $x=1+pw$. Then your equation becomes: 
$$ (1+pw)^n=1-p^{nN}a^n$$ 
(I forgot the $n$th power in the RHS) with $w\in \mathbb Z_p$. As $nN\ge 2r+1$, this equation has a solution. By varying $N$ or $a$, we get infinitely many solutions for $x^n+y^n=1$ with $x=1+pw$ and $y=p^Na$. 
