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Thread: give an example

  1. #1
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    give an example

    Prove that if $\displaystyle a^n -1 $ is prime where $\displaystyle a \in N $ and $\displaystyle n \geq 2, $ then $\displaystyle a =2 $ and $\displaystyle n $ is prime
    How I can use the identity : $\displaystyle a^{kl} -1 $=$\displaystyle (a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + …+$\displaystyle a^{k }+1) $ to solve this Q ??
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  2. #2
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    Quote Originally Posted by flower3 View Post
    Prove that if $\displaystyle a^n -1 $ is prime where $\displaystyle a \in N $ and $\displaystyle n \geq 2, $ then $\displaystyle a =2 $ and $\displaystyle n $ is prime
    How I can use the identity : $\displaystyle a^{kl} -1 $=$\displaystyle (a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + +$\displaystyle a^{k }+1) $ to solve this Q ??

    Well, if $\displaystyle n=kl$ is not a prime then, as you wrote, you have $\displaystyle a^{kl} -1= (a^k -1)(a^{k(l-1)} + a^{k(l-2)}$ + +$\displaystyle a^{k }+1) $ , and this is a non-trivial factorization of $\displaystyle a^n-1$

    UNLESS $\displaystyle a=2$ ...

    Tonio
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