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Find all positive integers n such that n^5 + n^4 + 1 is prime.

 Jun 18, 2020
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Notice that \(n^5 + n^4 + 1 = (n^2 + n + 1)(n^3 - n + 1)\).

 

That means, for all \(n^2 + n + 1 \neq 1\) and \(n^3 - n + 1 \neq 1\)\(n^5 + n^4 + 1\) is composite.

 

This means when \(n^5 + n^4 + 1\) is prime, \(n^2 + n = 0\) or \(n^3 = n\)

 

Solving, we get n = -1 (rejected) or n = 0 (rejected) or n = 1.

 

When n = 1, \(n^5 + n^4 + 1 = 3\), which is a prime.

 

The only possibility is n = 1.

 Jun 18, 2020

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