Knowing it's just 31 and 127 that were really new AND correct does it still seem like Mersenne truly "somehow figured out M(127)"? To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking of these particular values up until he felt confident enough to list them rather than rigorously prove them and it sticks out to us as M(127) is the highest of these that we retrospectively know to be valid.
> To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking
Of course the method was erroneous, that's not the point I am (apparently rather poorly) trying to get across. My point is that the method seems to have been lost to time, and that it is a shame because while faulty for finding primes it may prove fruitful elsewhere, or may simply be interesting. And if he had a method of checking for primes in the 17th century, this would also be noteworthy and interesting, though it would beg the question of why he stopped presumably before reaching a thousand.
My base observation is relatively straightforward and I honestly don't understand the dismissiveness it is met with:
Sometime in the early 1600s, Mersenne found M(127). The 'how' is interesting to me because I don't see it likely that this was discovered by random luck, but predicated on a novel method he had developed which, while clearly very flawed, would nonetheless still curious to see. Our math and science textbooks are filled with various flawed methods, why then is this one of seeming no concern?
Mersenne didn't "find" M(127) by using a method, he noted 2^n-1 was commonly prime for the first few well known values and wasn't able to immediately exclude M(127). That's the difference, not discovery by proof with some method rather lack of exclusion by method. Were the exclusions more reliable it'd be interesting but neither being able to exclude or include accurately is not.
The exact method doesn't matter because poorly performed naive division checking with the first couple thousand primes gets you a more accurate list of primes so whatever he was doing isn't particularly interesting (and could have been as simple as something like that that, we just can't say how he messed up for certain without a time machine).
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521
And Mersenne's was:
2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257
2-19 were proven naively prior to Mersenne with just 128 trial divisions leaving his portion of the list as:
31 (right), (missed), 67 (wrong), (missed), 127 (right) and 257 (wrong)
Knowing it's just 31 and 127 that were really new AND correct does it still seem like Mersenne truly "somehow figured out M(127)"? To me it seem like he either applied an erroneous method or, more likely in my opinion, just had erroneous naive checking of these particular values up until he felt confident enough to list them rather than rigorously prove them and it sticks out to us as M(127) is the highest of these that we retrospectively know to be valid.