The cumulative distribution of the gaps between values of Poisson process should be governed by an exponential distribution. Let's check the predicted Poisson gap distribution against the known Mersennes. One thing is that given one Mersenne prime exponent's p , the next one will fall, on the average, near 1.47576 p. Mersenne exponent might be about 10,000,000 yielding a Mersenne with about 3 million digits. Or it may not.
Lets be more specific graphically.
But we do know 38 Mersennes. So we created a separate page deriving the Wagstaff Mersenne conjectures .