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Show co-prime

  1. Sep 24, 2011 #1
    Show that if [itex]a,b\in\mathbb{N}^+,\ \gcd(a,b) = 1[/itex] and [itex]p[/itex] is an odd prime,
    then [itex]\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)\in \{1,p\}[/itex]
     
  2. jcsd
  3. Sep 25, 2011 #2
    Suppose [itex]1\le d \mid \gcd(a+b,\frac{a^p+b^p}{a+b})[/itex], then we have the following

    [itex]1\le d \mid a+b\implies b \equiv -a\ (\text{mod }d)\implies \sum_{k=0}^{p-1}(-1)^k a^{p-1-k}b^k \equiv pa^{p-1}(\text{mod }d)[/itex]

    [itex]\frac{a^p+b^p}{a+b} \equiv pa^{p-1}(\text{mod }d)[/itex]. Now since [itex]\gcd(d,a)=1[/itex], this means that [itex]d \mid p[/itex]
     
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