Hello, everybody. As usual I'm quoting different people. Disturbingly, I
noticed a weird HTML tag on my last E-mail. I can assure you I didn't put it
there, and don't know why my software's acting up on me (because I've never
seen that jibberish before). HTML mail is evil, only second to MIME. Long
live ASCII! Please excuse my delay in sending this message.
<<Also, if this extrapolation of the number of digits is accurate, there is
another prime between the 37th & 38th(p=6972593) discovered primes.
Unfortunately, the extrapolation of P just didn't go well. Actually, the
extrapolated 39th mersenne prime is 6.34% off of 2^6972593-2. I suppose
that's not so bad. That would also mean one was skipped. So it is currently
my fairly strong opinion that a mersenne prime was skipped between the 37th &
38th discovered primes. I reserve the right to change my mind at any
instant. I'd also guess that the skipped prime may have been pretty close to
2^5014947-1, and have a number of digits close to 1408773.>>
This is similar to the conjecture I made oh-so-long ago. 6.9M was confirmed,
but at the same time I had to predict a missing one (which I said was
probably in the 4M range, but that's far less certain).
<<[EMAIL PROTECTED]: I'm really looking forward to hearing how you made your
estimates.>>
<< How did you make this estimate ? Fit an exponential curve to the known
primes, and extrapolate the 1st one that should have at least 1m digits?>>
<<Who knows the mysterious ways of STL?>>
I usually just go by S.T.L. (my initials).
<<At some point I had a vague recollection that STL had believed there was a
number missing, and I was quite happy to see that it basically matched
what I got. And rounded, my estimate for #39 equals his. >>
Your estimates are close to what I have. I haven't the slightest idea how one
establishes priority for a discovery, but here I go. I think I've improved on
the conjecture of Wagstaff, Gillies, Lenstra, Pomerance, et al. (At the very
least, it's another conjecture.) Some time before I E-mailed the list with
my 3 conjectures I had made the main one. I had not told the list of it
because I planned to use it in a ridiculously important paper for school. But
it seems that my method is being rediscovered and I may as well announce what
I've seen. Here it is, and what led me to it.
I had picked up _Unsolved Problems in Number Theory, Second Edition_ by
Richard K. Guy, and was reading it. In section A3 (which contains a humorous
typo), I found the following:
<<Suppose M(x) is the number of primes p <= x for which 2^p - 1 is prime....
Lenstra, Pomerance, and Wagstaff all believe this [an early conjecture by
Gillies] and in fact suggest that ?? M(x) ~ e^gamma log x ?? where the log
is to base 2.>>
So I started investigating this. Armed with a copy of all Mersenne primes (I
have a bad habit of saying this when I really mean exponents for which the
Mersenne number is prime) from 2 to 3021377, a loyal TI-92+, and a newly
gained knowledge of basic statistics, I started computing some things.
First, I made a list of numbers 1-37. And I had the list of exponents, sorted
by size. (I called it mersenne.) Then I plotted points with the 1-37 values
for X and log[2] (Mersenne) for y. I got an astoundingly linear graph, as
others have.
Then, I plotted e^gamma log[2] (mersenne) versus the list of 1-37. Alongside
this I graphed y=x. This is because the y=x line represents the Wagstaff
conjecture. (If all exponents for which the corresponding Mersenne number is
prime followed the Wagstaff conjecture exactly, e^gamma log[2] (mersenne)
would equal 1, 2, 3, 4, 5, for each exponent in turn.) This was a rough
measure of how well the Wagstaff conjecture applies to the actual set of
Mersenne prime exponents. This graph seemed a little strange.
So, I graphed e^gamma log [2] (mersenne) - (1, 2, 3, 4, etc). This represents
how far off the Wagstaff conjecture is when applied to the data. (The
Wagstaff conjecture *should* say that M(3021377) = 37, but it doesn't. This
is why I graph this jibberish). This graph was INCREDIBLY disturbing. Save
for one Mersenne prime, all these "errors" were above 0, and often big. Ech!
So, I used my TI-92+ to take a linear regession line of this data (because I
had recently learned how to do regression lines and correlation
coefficients). This line was Y = .004769x + 1.4615. See what's happening
here? It seems that there's a consistent error (1.4615) in the Wagstaff
conjecture that doesn't change as the Mersenne primes grow (the .004769). So
I went back and applied this correction to the graph "that seemed a little
strange" and it fit y=x much better.
Hence, my new conjecture:
?? M(x) ~ e^gamma log[2] (x) + C ??
Of course, I used 1.4615 to make my 3 conjectures to the Mersenne mailing
list. In reality, I'm guessing it might be 1.5, or even 2^(1/e^gamma)! (In
fact, I'd rather go with 2^(1/e^gamma), as Erhardt chose 1.5 for the e^gamma
in the conjecture and now several mathematicians call the Erhardt Conjecture
"probably false") This is mildly disturbing, of course.
I think I haven't screwed up in formulating this conjecture (i.e. understood
the conjecture stated in Guy's book correctly). I know it's based on only
empirical evidence and no heuristic arguments. Please comment on it. When my
ridiculously important paper for school is done, I'll post it and give the
address to the Mersenne mailing list. (But I can't recieve help from you
guys. :-S ).
Thanks,
S.T.L.
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