1 | initial version |

Here is a possibility to work.

Let $A$ be a ring such that $2$ is invertible, for example $\Bbb Z/3$ or $\Bbb Z/9$. We denote by $\frac 12$ its inverse. Assume $2$ is not a square in $A$. (This is the case in the above examples.)

Consider $A[a]$ to be the ring $A[Y]/(Y^2-2)$, where $A[Y]$ is the polynomial ring in $Y$ over $A$. (And $a$ is $Y$ modulo the ideal generated by $(Y^2-2)$. So formally, $a=\sqrt2$.

Now observe that the equation satisfied by the $x$ element in the OP, $(x+a)^2=2(x+a)$, can be written as $$(x+a)(x+a-2)=0\ .$$ The two factors are relatively prime, in fact $$\frac 12(x+a)-\frac 12(x+a-2) = 1\ .$$ This means that the ring $$ R=A[a][X]\ /\ (\ (X+a)(X+a-2)\ ) $$ is isomorphic to two copies of $A[a]$ via the map $$ R\to A[a]\times A[a]\ ,\ f(X)\to(f(-a), f(2-a))\ . $$ The inverse map takes $(s,t)\in A[a]\times A[a]$ and maps it into $\frac 12t(X+a)-\frac 12s(X+a-2)$.

Now we have to solve $Z^2=1$ in the ring $A[a]\times A[a]$. This can be done also manually. An element of the shape $Z=u+va\in A[a]$, $u,v\in A$ satisfies $Z^2 = 1$ iff $(u^2+2v^2)+2auv=1$. In case $A$ has zero divisors we have to take care of $uv=0$ somehow. The possible values for $u,v$ that may lead to a solution satisfy at any rate $u^3=u$ and $2v^3=v$. Together with $Z$, also its "conjugate" $\bar Z=u-va$ is also a solution, and the "norm" $N(Z)=Z\bar Z=(u+va)(u-va)=u^2-2v^2$ is $1$. So it is a good idea to search for solutions of this "Pell equation" over $A$.

Let us now write some lines of code for the given case.

The brute force search is:

```
r = Zmod(9)
R.<a,x> = PolynomialRing(r, order='lex')
J = R.ideal( [a^2-2, (x+a)^2-2*(x+a)] )
S = R.quotient(J)
for [s, t, u] in cartesian_product([r, r, r]):
Z = S(s + t*a + u*x)
if Z^2 == S(1):
print(f"Z = {s} + {t} a + {u} x")
```

Results:

```
Z = 1 + 0 a + 0 x
Z = 1 + 8 a + 8 x
Z = 8 + 0 a + 0 x
Z = 8 + 1 a + 1 x
```

This fits with the situation of finding all points $Z=(Z_1,Z_2)$ over the ring $R=A[a]=\Bbb Z/9[a]=\Bbb Z/9[\sqrt 2]$ with $Z^2=(1,1)=1_R$.

```
sage: r = Integers(9)
sage: R.<Y> = r[]
sage: Q.<a> = R.quotient(Y^2-2)
sage: a^2
2
sage: for r1, r2 in cartesian_product([r, r]):
....: Z1 = r1 + r2*a
....: if Z1^2 == Q(1):
....: print(Z1)
....:
1
8
```

These are the only Hensel lifts of the corresponding units over $\Bbb Z/3$:

```
sage: U.<A> = PolynomialRing(GF(3))
sage: F.<a> = GF(3^2, modulus = A^2-2)
sage: a^2
2
sage: [ f for f in F if f^2 == F(1) ]
[2, 1]
sage: # of course, this is a field...
```

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