Mersenne numbers fail primality test at 2047 itself. How could we believe Mersennes are primes?

Question

M$_{11}=2047$ is a composite number. How could one, not check the primaility of such a small number and believe that all Mersenne numbers are primes?

Answer 1

I don't think anybody ever thought all Mersenne numbers are prime.

Wikipedia says:

Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257, as follows:

$$2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257$$

His list was completely accurate until 31, but then becomes largely incorrect, as Mersenne mistakenly included $M_{67}$ and $M_{257}$ (which are composite), and omitted $M_{61}$, $M_{89}$, and $M_{107}$ (which are prime). Mersenne gave little indication how he came up with his list

You may be confusing Mersenne numbers with the Fermat numbers, $2^{2^n}+1$. These were conjectured by Fermat to be prime, but Euler found that $2^{2^5}+1$ is composite.