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# Need help 2

$\gcd \left( a+b, \dfrac{a^p+b^p}{a+b}\right) = 1 \text{ or } p$

If $$p$$ is an odd prime and $$a,b$$ are co-prime positive integers, prove the equation above.

Note by Akshat Sharda
6 months, 2 weeks ago

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@Utsav Bhardwaj · 6 months, 2 weeks ago

@Otto Bretscher @Ivan Koswara · 6 months, 2 weeks ago

Nice one Akshat !!! @Akshat Sharda · 6 months, 2 weeks ago

Just an outline since I'm at work: You can write $$\frac{a^p+b^p}{a+b}=pa^{p-1}+(a+b)q(a,b)$$ for some polynomial $$q$$ (simple algebra exercise). Thus a common divisor of $$\frac{a^p+b^p}{a+b}$$ and $$a+b$$ is also a divisor of $$pa^{p-1}$$, which proves our point. · 6 months, 2 weeks ago