On Sun, 04 Nov 2001 21:46:58 -0500, Jud McCranie
<[EMAIL PROTECTED]> wrote:
>At 08:48 PM 11/4/2001 -0500, Nathan Russell wrote:
>
>>Of course, this whole argument makes (as far as I can see) heavy use
>>of the gamblers' fallacy, aka the fallacy of maturation of
>>probabilities ("Hey, I lost the last 50 games - what are the odds
>>against me losing 51 5-man games in a row? I'm certain to win!")
>
>Statistically, each exponent resulting in a prime is about twice the
>previous one. There is a heuristic argument that supports that being the
>case. The last one was 6.9 million something, we're now testing close to
>twice that. I think there are only three cases where one exponent is more
>than twice as large as the previous one.
P1 P2 ratio
127 521 4.102362205
607 1279 2.10708402
4423 9689 2.190594619
216091 756839 3.502408707
1398269 2976221 2.128503886
and, almost certainly,
3021377 6972593 2.307753385
It looks like there's some clumpiness around ratios of 2.1, but I
suspect that's an artifact of the cutoff point you gave. I wouldn't
know how to check statistically, and the sample size is probably too
small anyhow.
I don't know whether the second-largest gap is an outlier, but the
largest definately is. It might be noted that there are only
98-31-1=66 primes between 127 and 521 (exclusive); I don't know
offhand whether this is significant.
The gaps with ratio < 1.1 follow, for those who are curious.
4423 4253 1.039971785
9941 9689 1.026008876
21701 19937 1.088478708
23209 21701 1.069489885
3021377 2976221 1.01517226
I don't see any outliers here. I might note that gaps of this size
aren't even possible until after 29 (31/29=1.077, and 19/17=1.118,
thus excluding gaps beginning with the first seven Mersennes before
the primes above them are even calculated)
Any comments from the list?
Nathan
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