The BIG gap?

[R.D. Silverman]

Originally Posted by wblipp

An Erlang distribution is a sum of identical exponential distributions. The number in the sum is called the shape parameter or the stage parameter. So an exponential distribution is a particular case of the Erlang distribution, with 1 stage. The counting function for Erlang distributions with more stages is not Poisson.

Yes, I omitted that the distribution was a parameterized family.
My omission.

[davieddy]

Originally Posted by Mini-Geek

According to http://primes.utm.edu/notes/faq/NextMersenne.html, (I recommend you read that page, it has lots of details on this subject)

I agree with this recommendation with a few reservations.

Although Chris Caldwell acknowledges that not all exponents, below 43M
have been tested, he also suggests that the "expected" 1.5 gap should
be based on 43M.
As the OP is clearly aware of, the "starting point" is now ~50M.
What the next MP will be is slightly confused, because if
perchance it is found tomorrow, it would be anything from 43 to 53M.

When it will be found is slightly easier. Every 2+ days, the expected number of new primes before 79.3M (see the great "classic status" page)
decreases by 0.001. This suggests that the probability of no prime
before N days is e^(-0.001N/(2+)). (I'm sure William will correct this if necessary). 50% chance of a prime in 4 years from now.

This is consistent with your observation of a 3/50 (6%) advance of the
LL wave per year which requires a 20% increase in computing power per year. Not too much to ask of Moore's Law one would hope.

Re Caldwell's predictions of when a 100M or 1000M digit prime be found, note that on the log scale used, the GIMPS discoveries suffer from "Brewer's Droop" when compared to his rigid rod

David

PS Welcome to the House of Fun.