number theory - Modular equations and congruences

I have the following question, actually from the book Rational Points on Elliptic Curves by Silverman and Tate;

Prove that for every exponent $e \geq 1$, the congruence
$$x^2 +1 \equiv 0 (\text{mod } 5^e)$$

has a solution $x_e \in \mathbb{Z} / 5^e \mathbb{Z}$. Prove further that these solutions can be chosen to satisfy

$$x_1 \equiv 2 (\text{mod } 5),\qquad \text{and}\quad x_{e+1} \equiv x_e (\text{mod } 5^e) \quad \text{for all } e \geq 1.$$

The best way to go about it is supposed to be induction on $e$. So I have gone through the initial step, i.e. for $e=1$ and shown that there is indeed a solution to the congruence in $\mathbb{Z} / 5^e \mathbb{Z}$, namely $x=2$ or $x=3$. However my problems begin from here on out. I just don't know how to proceed with this problem. This question is actually already (Prove that for every exponent $e\ge 1$, the congruence $x^2+1\equiv 0$ (mod $5^e$) has a solution $x_e\in \mathbb{Z}/5^e\mathbb{Z}$). I just can't follow the answer at all and was hoping someone could provide a more basic answer or methodology I could follow. Thank you for your help!

Answer

Starting with $x_1\equiv 2 \mod 5$ and $x_2\equiv x_1 \mod 5$, we have $x_2=5k+2$ for some integer k. Then

$x_2^2+1 = 25k^2 + 20k + 5 \equiv 20k+5 \mod 25$

So $k \equiv 1 \mod 5$ and $x_2 \equiv 7 \mod 25$

Then wash, rinse and repeat ...

$x_3\equiv x_2 \mod 25$

$\Rightarrow x_3 = 25k + 7$

$\Rightarrow x_3^2 + 1 = 625k^2 + 350k + 50 \equiv 100k + 50 \mod 125$

$\Rightarrow k \equiv 2 \mod 5$

$\Rightarrow x_3 \equiv 57 \mod 125$

etc.

The general case is known as Hensel's lemma.