He did not just stumble onto two correct primes in the 17th century by chance.
M31 you can verify by hand, which really leaves us with one out of four. It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold. There are plenty of neat-looking lines in, say, an Ulam spiral but those don't generate primes either.
> It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold
This is exactly the attitude I alluded to in my earlier post. You think M(127) was found arbitrarily through chance or plucked from a likewise arbitrary pattern, which then allows you to completely write him off...as opposed to the possibility I'm asking about: that he actually did have an interesting method of analyzing primes which, while faulty, could nonetheless prove fruitful or even just interesting. It's this dismissiveness that I don't understand.
I guess I don't see the 'dismissiveness' bit. You have some hunch that there was something deeper to Mersenne's noodling. That's perfectly fine but I think both the sibling commenter and I have laid out a decent argument that, if you poke around the hunch a bit, there doesn't seem to be much there. The pattern, such as it is, is no good at identifying primes and even worse at picking out what we now call Mersenne primes. I'm not sure I follow what your argument is. It doesn't take much mathleticism to notice he picked four exponents close to doubling powers of two. The link in Wikipedia goes into a bit of detail:
Which seems both plausible, or at least in the ballpark and not terribly mysterious. He got a big number right. It happens.
That's not to say there aren't mathematical mysteries and curiosities of that nature. You have to look no further than Mersenne's contemporary and correspondent Pierre de Fermat for a solid one. He cranked out a few dozen theorems with next to nothing in the way of surviving proof. From the famous one on down, he got just about all of them right. People have been wondering aloud how exactly he pulled that off since at least Gauss's time. Nearly as famously, he got one wrong:
These look, in form, reminiscent of Mersenne's numbers and the first few of them are also prime. F5 is not. In a fun connection, Euler, the same dude who showed M31 is prime showed F5 is not. You can almost imagine Mersenne and Fermat shaking their fists and yelling 'Euleeeeeeer!' Seinfeld-like, from their graves.
M31 you can verify by hand, which really leaves us with one out of four. It's entirely possible he got one right by chance. It probably wasn't strictly by chance, very likely convinced himself of some pattern that doesn't really hold. There are plenty of neat-looking lines in, say, an Ulam spiral but those don't generate primes either.