A very nice Q

prove that if 2ⁿ - 1 is prime then n is also prime

5 Answers

i think u wont need the proof MI.i dono the elegant proof.

trying to do with binomial coeff's

Contradiction is the best method.

Let n be composite.
So it can be written as n = x.y

2^{n}-1 = 2^{x.y}-1 = (2^{x}-1).(1+2^{x}+2^{2x}+...+2^{(y-1).x})

This makes 2^{n}-1 a composite no. whereas it is PRIME.

Hence, n has to be prime when 2ⁿ - 1 is prime (but the converse is not always true).

well done
rkrish

Just quoting an well-known result....

If a^k-1 is a prime, then a=2, and k is a prime.....

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