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Let $p$ be prime number ($p\gt2$) and $a,b\in\mathbb Z$ ,$a+b\neq0$ ,$\gcd(a,b)=1$ how to prove that $$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1~~\text{or}~~ p$$

Thanks in advance .

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See also…, and other questions linked there. – Martin Sleziak Mar 22 '14 at 7:04

1 Answer 1

up vote 7 down vote accepted

$\rm\displaystyle \left(a\!-\!b,\frac{a^p\!-b^p}{a-b}\right) = (a\!-\!b,\,a^{p-1}\!+\cdots+b^{p-1}) = (a\!-\!b,\color{#C00}{pb^{p-1}}) = (a\!-\!b,p)\:$ by $\rm\:a\equiv b\ \ (mod\ a\!-\!b)\:$ and $\rm\:(a,b) = 1\:\Rightarrow\: (a\!-\!b,b) = (a,b)=1\ \cdots\:\Rightarrow\:(a\!-\!b,b^n) = 1\ $ by Euclid's Lemma.

Remark $\ $ It's a special case $\rm\ f(x) = x^p,\,\ x = a\ \ (so\ \ \color{#0A0}{f'(b)} = \color{#C00}{pb^{p-1}})\:$ of

Theorem $\rm\displaystyle\quad \frac{f(x)-f(b)}{x-b} \: \equiv\ \color{#0A0}{f\:'(b)}\ \ \ (mod\ \:x\!-\!b)\quad$ for $\rm\ f(x)\in \mathbb Z[x]$

Proof $\ $ By Taylor, $\rm\,\ f(x)\ =\ f(b) +\: f\:'(b)\ (x\!-\!b) \,+\, (x\!-\!b)^2\: g(x)\ \ $ for some $\rm\ g(x) \in \mathbb Z[x]$

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Note: this is meant as a (big) hint, along with some conceptual background for further motivation. If you have any trouble filling in the details then please feel welcome to ask questions in comments, and I will gladly elaborate. – Math Gems Mar 25 '13 at 20:05
See also this answer. – Bill Dubuque Apr 22 at 15:26

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