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If $\gcd(a,10)=1$ and $n\ge 1$, is the equation $$a^x\equiv x\mod 10^n$$ always uniquely solveable modulo $10^n$ ? If yes, how can this be proven ?

The discrete logarithm does not seem to help. I checked that there is always a unique solution for $2\le a\le 100$ and $1\le n\le 7$. I know that this is related to the power towers $a\uparrow a\uparrow a\cdots $ (and this was discussed several times on this site), but I cannot remember a proof that it always works.

Additional question : Does the sequence $x_1=a$ , $x_{n+1}=a^{x_n}$ always get stationary modulo $10^m$ with $m\ge 1$ (that means $x_{n+1}\equiv x_n\mod 10^m$) , if $\gcd(a,10)=1$, and can this be proven as well ? It seems that the sequence becomes stationary at index $m+1$.

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