Mersenne primes and exponents

#Mersenne#primes#exponents

Statement

Show that if $2^n-1$ is prime, then $n$ is prime.

Hint

If $n=ab$ , $2^n-1=(2^a)^b-1$ factors.

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Factorization. $2^{ab}-1=(2^a-1)((2^a)^{b-1}+\cdots+1)$ .

Contradiction. If $a,b>1$ , both factors are $>1$ , so $2^n-1$ composite.

Conclusion. $n$ must be prime.

Example. $2^4-1=15$ composite; $n=4=2\cdot2$ not prime. ✓