Waste less time on Facebook — follow Brilliant.
×

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

No vote yet
1 vote

Comments

Sort by:

Top Newest

@Utsav Bhardwaj Lakshya Sinha · 6 months, 2 weeks ago

Log in to reply

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

Log in to reply

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

Log in to reply

@Akshat Sharda 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. Otto Bretscher · 6 months, 2 weeks ago

Log in to reply

@Otto Bretscher Exactly !!!DId the same ! Chinmay Sangawadekar · 6 months, 2 weeks ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...