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Let $n,a,b \in \mathbb Z^+$ be such that $n\mid a^n-b^n$ , then how to prove that $n\mid {\dfrac {a^n-b^n}{a-b}}$ ?

My try :

$d=\gcd(n,a-b),$ so $d \mid{\dfrac {a^n-b^n}{a-b}}.$ Also $\,n \mid(a-b)\left({\dfrac {a^n-b^n}{a-b}}\right),\;$ so $\;\dfrac nd\mid\dfrac {(a-b)}d\Big({\dfrac {a^n-b^n}{a-b}}\Big)$.

Now, since $\gcd\left(\dfrac nd ,\dfrac {(a-b)}d \right)=1,\;$ so $\dfrac nd\mid{\dfrac {a^n-b^n}{a-b}},\;$ it follows that $\;\operatorname{lcm}\left(d, \dfrac nd \right)\mid {\dfrac {a^n-b^n}{a-b}}.$

But then I am stuck , Please help . Thanks in advance

amWhy
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1 Answers1

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Hint: First do the case where $n$ is a prime power.

Further hint: By induction on $k$, if $p$ is prime, then $p^m\mid a-b$ implies $p^{m+k}\mid a^{p^k}-b^{p^k}$ for all $k$. (Use binomial expansion in the induction step.)

Then conclude for composite $n$.

Further hint: If $p^k\mid n$, write $\frac{a^n-b^n}{a-b}=\frac{a^n-b^n}{a^{n/p^k}-b^{n/p^k}}\cdot\frac{a^{n/p^k}-b^{n/p^k}}{a-b}$

Bart Michels
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