Proving (n-1)|(n^k - 1) and the Primality of n^k - 1 when n=2 and k is Prime

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Homework Help Overview

The problem involves proving that (n-1) divides (n^k - 1) for integers n and k, where n is at least 2 and k is at least 2. Additionally, it seeks to establish that if (n^k - 1) is prime, then n must equal 2 and k must be prime.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster considers using proof by induction as a potential approach but expresses uncertainty about how to proceed. Some participants provide a hint involving a factorization of (n^k - 1), while others clarify the notation used in the problem statement.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem and clarifying the expression involved. A hint has been provided, but there is no explicit consensus or resolution yet.

Contextual Notes

There is a focus on the correct interpretation of the expression (n^k - 1), with some participants questioning the notation used in the original post.

Fairy111
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Homework Statement



Let n and k be integers with n>=2 and k>=2. Prove that (n-1)|(n^k - 1).
Hence prove that if n^k - 1 is prime then n=2 and k is prime.

Homework Equations





The Attempt at a Solution



I think you go about this question by using proof by induction. However I am really not sure how to do this. Any help would be great! Thanks
 
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Hint: [tex](n-1)(n^{k-1}+n^{k-2}+\ldots+n+1)=n^k-1[/tex]
 
its supposed to be n^(k) - 1
 
Fairy111 said:
its supposed to be n^(k) - 1

Isn't that the same as the RHS of the formula I wrote?
 

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