Okay, so I got my line of log base 2 of the exponents of the 1st 37
mersenne primes. I took those numbers & did a linear extrapolation, and
did a 2^n to the resulting extrapolated numbers.
I then went back and did my exponential extrapolation to the exponents of
the 1st 37 primes.
I was pretty surprised that the extrapolations for M38-M42 (all that I
did) were *exactly* the same for both methods of extrapolations, based on
2 different fitted lines (one exponential based on the exponents, and one
linear, based on the log base 2 of the exponents).
If you care, M38-M42 came out as:
#38 5014947.208
#39 7414614.414
#40 10962529.54
#41 16208132.64
#42 23963772.48
The (linear) line fitted to the log base 2 of the exponents was
y = 0.5641x + 0.8206 with R2 = 0.9925
The (exponential) line fitted to the exponents was
y = 1.7661e^(0.391x) with R2 = 0.9925
Hmm... R2's match. So they are of equal use.
I still wanna know why extrapolating off of the number of digits, instead
of the actual exponents, gave me a number closer to 6972593 (38th
discovered mersenne prime). I dunno, coulda just been a coincidence.
I'm trying to find out what exactly the lines are that are conjectured to
fit this data.. like on
http://www.utm.edu/research/primes/notes/faq/NextMersenne.html -- gives
the slope, but I haven't found the offset. I want to see how reliably
that can be used to predict these numbers (like, taking off the 37th, &
predicting w/ 1-36).
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