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Ok, so I know when $(a, m) = 1$, by Euler's Theorem, $a^{\phi(m)} \equiv 1 \mod m$. Since $\phi(323) = 288$, $a^{288} \equiv 1 \mod m$ when $(a, 323) = 1$. However, there are some elements $a$ such that $(a, 323) \not= 1$ and $a^{288} \not\equiv 1 \mod 323$. Since those elements do not have multiplicative inverses in $\mathbb{Z}/323$, how is it working that $x^n$ is invertible? Am I missing something?

Exercise I.8. Prove that $f : \mathbb{Z}/323 \to \mathbb{Z}/323$ given by $f(x) = x^n$ is a bijective map if $(n, 6) = 1$.

Proof. Assume that $(n, 6) = 1$. Then $2 \not\mid n$ and $3 \not\mid n$. Let $f(x) = x^n$. Now by Theorem 9.3, $\phi(323) = \phi(17 \cdot 19) = (17-1)(19-1) = 16 \cdot 18 = 288 = 2^5 \cdot 3^2$. We need $x$ such that $nx \equiv 1 \mod 288$. Since $2 \not\mid n$ and $ \not\mid n$, $(n, 288) = (n, 2^5 \cdot 3^2) = 1$. Then by Lemma 5.2, $nx \equiv 1 \mod 288$ has exactly one solution. That is, $n^{-1}$ exists in $\mathbb{Z}/288$. Then $f^{-1} = x^{n^{-1}}$ since $f^{-1}(f(x)) = f^{-1}(x^n) = (x^n)^{n^{-1}} = x^{n \cdot n^{-1}} \equiv x \mod 323$. Since $f$ is invertible, $f$ is bijective. $\blacksquare$

(Image version)

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Joseph DiNatale
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2 Answers2

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It works here because $\,323 = 17\cdot 19\,$ is squarefree, so the following generalization of Euler-Fermat implies that if $\rm\,{\rm lcm}(16,18) = 144\mid e-1\,$ then $\, a^e\equiv a\pmod{323}\,$ for all $\,a.$ This fails if the modulus $\rm\,m = k d^2,\, d>1\,$ is not squarefree, since then $\rm\,(kd)^e \equiv 0^e \pmod{m}$ for all $\rm\,e\ge 2\,$ therefore the map $\rm\,f(x) = x^e\,$ is not $\,1$-$1,\,$ since $\rm\,f(kd)\equiv 0 \equiv f(0)\,$ but $\rm\,kd\not\equiv 0.$

Theorem $\ $ For natural numbers $\rm\:a,e,n\:$ with $\rm\:e,n>1$

$\qquad\rm n\:|\:a^e-a\:$ for all $\rm\:a\:\iff n\:$ is squarefree, and prime $\rm\:p\:|\:n\:\Rightarrow\: p\!-\!1\:|\:e\!-\!1$

Proof $\ (\Leftarrow)\ \ $ Since a squarefree natural divides another iff all its prime factors do, we need only show $\rm\:p\:|\:a^e\!-\!a\:$ for each prime $\rm\:p\:|\:n,\:$ or, that $\rm\:a \not\equiv 0\:\Rightarrow\: a^{e-1} \equiv 1\pmod p,\:$ which, since $\rm\:p\!-\!1\:|\:e\!-\!1,\:$ follows from $\rm\:a \not\equiv 0\:$ $\Rightarrow$ $\rm\: a^{p-1} \equiv 1 \pmod p,\:$ by little Fermat.
$(\Rightarrow)\ \ $ See this answer

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Bill Dubuque
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Note that $f$ being invertible doesn't mean that all elements of $\mathbb{Z} / 323\mathbb{Z}$ are invertible. That $f$ is an invertible map just means that there is an inverse map $g$ such that $$\begin{align} f \circ g &= 1 \\ g \circ f &= 1. \end{align} $$ That is: $f(g(x)) = x$ for all $x \in \mathbb{Z} / 323\mathbb{Z}$ and $g(f(x)) = x$ for all $\mathbb{Z} / 323\mathbb{Z}$. When we have such a $g$, we usually denote it by $f^{-1}$. So in your case $f(x) = x^{n^{-1}}$ where $n^{-1}$ is the inverse of $n$ in $\mathbb{Z} / 288\mathbb{Z}$

This relies in part on $n$ satisfying $(n, 6) = 1$.

Thomas
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