[QUOTE=NBtarheel_33;394119]0.497 now, so (with the standard caveat about calculations of this nature over extremely short intervals) that's a drop of 0.003 in six days, or basically 2,000 days (~5.5 years) until we expect to find a new prime.[/QUOTE]
To me that sounds a bit fishy. How did you calculate it?
Are you sure you're not just estimating the time until all exponents under 79.3M have been tested?

[QUOTE=NBtarheel_33;394119]0.497 now, so (with the standard caveat about calculations of this nature over extremely short intervals) that's a drop of 0.003 in six days, or basically 2,000 days (~5.5 years) until we expect to find a new prime.[/QUOTE]
It 2014 it dropped by 0.205 . Use that figure instead. I can give you a lot more data to play with if you want.

[QUOTE=Uncwilly;394138]It 2014 it dropped by 0.205 . Use that figure instead. I can give you a lot more data to play with if you want.[/QUOTE]

That would give 365/0.205 = ~1,780 days, or just under 5 years. Back when Davieddy was stalking the forum, he was consistently getting right around 4 years. So either GIMPS is slowing down (possible but the throughput numbers seem to say differently), or (more likely) we are seeing the effects of (1) the DC tail being cut down to size and (2) the greater computational expense of LL tests in the highest part of the "classical" range.

[QUOTE=NBtarheel_33;394211]That would give 365/0.205 = ~1,780 days, or just under 5 years.[/QUOTE]
If you mean the expected time until the discovery of a new prime, please could you justify it for the benefit of those of us who are interested but don't understand?:smile:

[QUOTE=Brian-E;394226]If you mean the expected time until the discovery of a new prime, please could you justify it for the benefit of those of us who are interested but don't understand?:smile:[/QUOTE]

First of all, keep in mind that in these calculations, we are making the [B]huge[/B] assumption that the distribution of Mersenne primes obeys a Poisson distribution. We have no proof that this is the case, but heuristics indicate that it is plausible.

Assuming a Poisson distribution, we can then calculate the probability of finding a prime in a given interval of exponents (e.g. between 2 and 79,300,000, the classical GIMPS upper limit; or between 50,000,000 and 60,000,000, etc.). From this, we can calculate the number of primes that we might expect to find in such an interval. Right now, for instance, between exponents 2 and 79,300,000, we presently expect to find 0.496 primes, per [URL="http://www.mersenne.org/report_classic/"]this[/URL] report. (Keep in mind that this is making that all-important, nontrivial, unproven assumption that the distribution of Mersenne primes is a Poisson process!)

Well, from the change in this expected number of primes over a time interval, we can estimate the time interval in which we would expect to find exactly one prime. This is a calculation that Davieddy would frequently make and from which he would infer the increase (or decrease) in GIMPS throughput (whether this is a valid metric for measuring GIMPS throughput is another argument for another time). Anyway, the logic is as follows: If the expected number of primes in an interval decreases by some amount [TEX]dE[/TEX] over a time interval [TEX]dT[/TEX], then [TEX]\frac{dE}{dT}[/TEX] roughly approximates the expected number of primes per unit of time (usually we measure [TEX]dT[/TEX] in days). Once we know the expected number of primes per unit of time, we can flip this around to ask the question: How long before the expected number of primes is exactly one?

To calculate this, we simply invert [TEX]\frac{dE}{dT}[/TEX] (which gives us primes per unit time interval) and calculate [TEX]\frac{dT}{dE}[/TEX] (which gives us time intervals per unit prime (i.e. how long for the expected number of primes to drop by exactly one). This quantity tells us (assuming the Poisson distribution holds!) the length of time from right now that we should expect to wait before we find a new prime. Note that, as Davieddy often remarked in the past, it doesn't matter how long we've *been* waiting; this figure tells us how much longer we should [B]expect[/B] to wait. So, even if we haven't found a new prime in, say, twenty years, if the figure [TEX]\frac{dT}{dE} = 20[/TEX] years (Heaven forbid!), according to Poisson, we should expect another twenty years of waiting.

Finally, an example, just to make things as clear as mud. As Uncwilly posted upthread, the expected number of primes dropped by 0.205 in 2014. This gives [TEX]dE = 0.205[/TEX] and [TEX]dT = 365[/TEX] (the length of the time interval is 365 days, i.e. the entire year 2014). We first note that this means that we should expect to find [TEX]\frac{dE}{dT} = \frac{0.205}{365} = 0.00056[/TEX] new primes per day. On the other hand, this also means that we should expect it to be [TEX]\frac{dT}{dE} = \frac{1}{\frac{dE}{dT}} = \frac{1}{0.00056} = ~1,786[/TEX] days [B]from the end of 2014[/B] (i.e. November 21, 2019) before we find a new prime. (Assuming that the Mersenne primes are distributed in a way that obeys the Poisson distribution!)

Hope this helps. Let me know if you still have questions. :smile:

[QUOTE=NBtarheel_33;394262]Hope this helps. Let me know if you still have questions. :smile:[/QUOTE]
Thanks very much for the very detailed description. What I didn't understand, but I think I do now thanks to your explanation, is how it was justified to use the decrease in the expected number of primes in a particular arbitrary interval (up to exponent 79.3M in the example). But now I see that you use a gradient (rate of decrease) of expected number of primes over that same interval and assume that it will apply over any larger interval (Poisson distribution).

Thanks for your patience with me.

Only 44 Mersenne primes below 10M digits
 

And as expected the [url=http://www.mersenne.org/report_exponent/?exp_lo=33185861&full=1]last of the <10M digit exponents[/url] was indeed finished by someone other than the registered [strike]slowcoach[/strike] user. This time by "Mike Neurohr".

[QUOTE=retina;394304]And as expected the [url=http://www.mersenne.org/report_exponent/?exp_lo=33185861&full=1]last of the <10M digit exponents[/url] was indeed finished by someone other than the registered [strike]slowcoach[/strike] user. This time by "Mike Neurohr".[/QUOTE]

This looks like it might very likely have been a legitimate "recycling" by Primenet, rather than a "poaching" (my spidering is not at a high enough temporal resolution to be able to say for sure).

[CODE]20141115 33185861 D LL, 54.90% 143 26 2014-06-25 2014-11-14 2014-11-15 2014-12-11 nranks
20141123 33185861 D LL, 54.70% 151 27 2014-06-25 2014-11-23 2014-11-24 2014-12-20 nranks
20141210 33185861 D LL, 53.50% 168 28 2014-06-25 2014-12-09 2014-12-10 2015-01-07 nranks
20150101 33185861 D LL, 58.20% 221 18 2014-06-25 2015-01-30 2015-01-31 2015-02-19 nranks[/CODE]

Assigned to "nranks" under the new recycling rules (possibly as a "Cat 2"); gratiously given over 220 days to complete. Making very slow progress.

[QUOTE=chalsall;394320]This looks like it might very likely have been a legitimate "recycling" by Primenet, rather than a "poaching" (my spidering is not at a high enough temporal resolution to be able to say for sure).[/QUOTE]I doubt that "Mike Neurohr" could get the assignment and then finish the test in 0.0 days. The Recent Cleared report was very clear about the timespan of 0.0 days.

[QUOTE=retina;394322]I doubt that "Mike Neurohr" could get the assignment and then finish the test in 0.0 days. The Recent Cleared report was very clear about the timespan of 0.0 days.[/QUOTE]

As I said, I don't have enough information to say for sure. But I can tell you that many of my machines can clear two DCs in less than 24 hours.

[QUOTE=chalsall;394323]As I said, I don't have enough information to say for sure. But I can tell you that many of my machines can clear two DCs in less than 24 hours.[/QUOTE]I'm not sure about the rounding used but I suppose at the most it could be 0.099999... days, i.e. just under 2.4 hours. Is such a time possible with current technology?