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Math Help - Proof of Prime number

  1. #1
    Junior Member
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    Proof of Prime number

    Hello, could anyone give me some hints to do these following questions? Any help will be greatly appreciated, thanks

    a) Let a and n be integers greater than 1. Prove that a^n -1 is prime only if a = 2 and n is prime.

    b) Show that 2^n + 1 is prime only if n is a power of 2.
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  2. #2
    Super Member PaulRS's Avatar
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    a) a^n-1=(a-1)\cdot{\left(1+a+...+a^{n-1}\right)} so what happens if a-1>1 ? (use this same identity-changing a- to prove that n has to be prime)

    b) note that <br />
a^{2k + 1}  + 1 = \left( {a + 1} \right) \cdot \left( {1 - a + a^2  \mp ... + a^{2k} } \right)<br />
now suppose that n has an odd divisor d and write <br />
\left( {2^{\tfrac{n}<br />
{d}} } \right)^d  + 1 = 2^n  + 1<br />
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