Finding a $x$ s.t $x^e \equiv y \pmod{p^2}$ given $x_1^e \equiv y\pmod p$

Question

I've a question that asks me to find an $x$ such that $$ x^e \equiv y \mod p^2 $$ and I know that $$ x_1^e \equiv y \mod p $$ where $1 \leq x_1 \leq p-1$ and $e$ is a positive integer.

My attempt

We have $ y = x_1^e -pk $ for a fixed $k$.

Also, we know that $ y = x^e - pk' $ for an arbitrary $k'$.

Hence, we can compute $x$ as $$ x = \sqrt[e]{x_1^e + p(k - pk')}$$ by giving an arbitrary integer to $k'$.

I feel my solution is wrong but can't really see why. Any help is much appreciated.

Answer 1

Suppose $x_1^e\equiv y+pk\bmod p^2$ then we have $(x_1+pj)^e \equiv x_1^e+x_1^{e-1}pj\bmod p^2$ (use newton's theorem and notice most summand vanish).

This is congruent to $y+pk+x_1^{e-1}pj\equiv y+(k+x_1^{e-1}j)$p.

So you just need to pick $j$ such that $x^{e-1}j\equiv -k\bmod p$ in other words $j\equiv -k x_1^{-e+1}\bmod p$.