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Prove that \(5^{3^n}+1\) is divisible by \(3^{n+1}\) for all nonnegative integers \(n.\)

 

What I did (or tried to do) was to use induction. I proved it worked for the base case \(0,\) but I'm stuck on how to prove it works for \(n=k+1\) if we assume it works for \(n=k.\) Help!

 

--------Thanks! laughcoolsmiley

 May 5, 2020
 #1
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Have you tried the sum of cubes formula? Try expressing the left side term with the sum of cubes factorization(the 5 term and 1).

 May 5, 2020
edited by jfan17  May 5, 2020
 #2
avatar+31316 
+6

Here's a proof by induction:

 

 May 5, 2020
 #3
avatar+279 
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WOW! Thanks Alan! laugh

madyl  May 5, 2020

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