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

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: https://primes.utm.edu/glossary/page.php?sort=MersennesConjecture https://primes.utm.edu/glossary/page.php?sort=MersennesConje... 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: https://en.wikipedia.org/wiki/Fermat_number https://en.wikipedia.org/wiki/Fermat_number 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.