On ring $\mathbb{Z}_{128}$, solve each of the following equations: $$(i)~x^2= x,~~~~~~(ii)~x^2=1,~~~~~~(iii)~x^{32}=1.$$

Attempt. Some thoughts.

(i) Clearly $x=0,1$ are solutions. Let $x \in \mathbb{Z}_{128}$ be such that $x^2=x$. Then $128=2^7$ divides $x^2-x=x(x-1)$.

(ii) Clearly $x=1,-1$ are solutions. Let $x \in \mathbb{Z}_{128}$ be such that $x^2=1$. So $xx=1$ and $x$ is invertible in $\mathbb{Z}_{128}$. So $\gcd(x,2^7)=1$ and $x$ is odd. Also, the order of $x$ is $2$ (unless, of course, the case $x=1$, which clearly satisfies the equation).

(iii) Clearly $x=1,-1$ are solutions. Let $x \in \mathbb{Z}_{128}$ be such that $x^{32}=1$, meaning that $x$ is invertible in $\mathbb{Z}_{128}$ and $x$ is odd and also the order of $x$ divides $32$.

Thanks for the help.

  • $\begingroup$ $x=\pm 63$ also solves (ii) $\endgroup$ – J. W. Tanner Feb 10 at 16:17

(i) You're almost there! What can you tell about $x$ given that $2^7$ divides $x(x-1)$? Note that it is impossible for both $x$ and $x-1$ to be even.

(ii) This time observe $x^2-1=(x+1)(x-1)$. Continue as you do in (i).

(iii) If $x^{32}=1$, then $x^{16}$ satisfies (ii). We can use this to find what $x^8$ must be, what $x^4$ must be, what $x^2$ must be, and then what $x$ must be; it might look like this would require an exponential amount of calculation, but much computation is saved by observing $x^2\equiv 1\pmod{4}$ if $x$ is odd.


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