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if 2^n+1 is prime then prove that n isa power of 2

if 2^n+1 is prime then prove that n isa power of 2

Grade:12

1 Answers

Sher Mohammad IIT Delhi
askIITians Faculty 174 Points
7 years ago

a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^kb^{n-1-k}.

Ifnis a positive integer but not a power of 2, thenn = rswhere1 \le r < n,1 <  s \le nandsis odd.

By the preceding lemma, for positive integerm,

(a-b) \mid (a^m-b^m)

where \mid means "evenly divides". Substitutinga = 2^r,b = -1, andm = sand using that s is odd,

 (2^r+1) \mid (2^{rs}+1),

and thus

 (2^r+1) \mid (2^n+1).

Because1 < 2^r+1 < 2^n+1, it follows that2^n+1is not prime. Therefore, by contraptionnmust be a power of 2.


sher mohammad

b.tech, iit delhi

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