Solutions of the congruence $x^2 \equiv 1 \pmod{m}$

For $m>2$, if a primitive root modulo $m$ exists, prove that the only solutions of the congruence $x^2 \equiv 1 \pmod m$ are $x \equiv 1 \pmod m$ and $x \equiv -1 \pmod m$.

Thanks.

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Here is an indirect solution – a solution that doesn’t use any specific primitive root. I think a better solution exists, but I would still like to post it because it gives a different perspective.

The solutions $1$ and $m-1$ are obviously always solutions. So the difficulty is proving there aren’t any more.

By the primitive root theorem, the possibilities for $m$ are $m=2$, $m=4$, $m=p^k$, or $m=2p^k$ where $p$ is an odd prime and $k\ge 1$.

For $m=2$ we actually have $1=m-1$, and there is a unique square root of $1$.

For $m=4$ you may check directly that the claim holds, just by calculating.

For $m=p^k$, we prove by induction on $k$. For $k=1$ there are no zero divisors, so $(x-1)(x+1)=0$ implies $x-1=0$ or $x+1=0$. For $k>1$, assuming there are exactly two solutions modulo $p^{k-1}$, we use Hensel’s lemma to claim that these lift to exactly two solutions modulo $p^k$.

For $m=2p^k$ we use the Chinese remainder theorem to combine the two solutions modulo $p^k$ with the unique solution modulo $2$ to get two solutions modulo $m$.

$x^2$$\equiv$1 mod m means that m|($x^2$-1)…..what can you infer from this?

Here is a direct solution involving a primitive root (see also my other answer for an “indirect” solution):

Let $r$ be a primitive root modulo $m$ and let $\{r^0, r^1, \ldots, r^s\}$ be the multiplicative group mod $m$.

Let $x$ satisfy $x^2 = 1$. Then $x$ must be a member of the multiplicative group, since it’s invertible mod $m$ (it is its own inverse). So we can write $x = r^k$ for some integer $k$, and we know that $r^{2k} \equiv r^0 \pmod m$. This gives us $2k \equiv 0 \pmod {\phi(m)}$ where $\phi$ is Euler’s totient function. This congruence has at most two solutions (this I leave as exercise), giving at most two solutions for $x$.

Suppose $\,w\,$ is a primitive root modulo $\,m\,$ , and $\,x=w^k\,$ , then

$$x^2=1\pmod m\iff w^{2k}=1\pmod m\iff$$

$$ (w^k-1)(w^k+1)=0\pmod m\iff (x-1)(x+1)=0\pmod m\ldots$$

The fact that any unit $\,x\,$ modulo $\,m\,$ is a power of $\,w\,$ follows from the fact of the latter being a primitive root modulo $\,m\,$…If $\,m\,$ has no primitive roots then the claim isn’t true.

Warning: In the last step you still have to give a very little explanation…

hint: an equation of degree 2, also in modular arithmetic, has at most two solutions if a primitive root modulo $m$ exists