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Math Help - A divisibility problem

  1. #1
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    A divisibility problem

    Show that if (a,b)=1 and p is an odd prime, then

    ( a+b, (a^p+b^p)/a+b ) = 1 or p
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  2. #2
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    Quote Originally Posted by eyke View Post
    Show that if (a,b)=1 and p is an odd prime, then

    ( a+b, (a^p+b^p)/a+b ) = 1 or p
    \frac{a^p+b^p}{a+b}=\sum_{j=1}^p (-1)^{j-1} a^{p-j}b^{j-1}= \sum_{j=1}^p (-1)^{j-1} (a+b \ - \ b)^{p-j}b^{j-1} \equiv pb^{p-1} \mod a+b. similarly \frac{a^p+b^p}{a+b} \equiv pa^{p-1} \mod a+b. so if d \mid a+b and d \mid \frac{a^p + b^p}{a+b}, then d \mid pa^{p-1} and d \mid pb^{p-1}.

    thus d \mid p \gcd(a^{p-1},b^{p-1})=p. \ \Box
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