Mersenne primes have the form 2p -1 where p is a prime. The graph below plots the trend in the size of these numbers based on the
50 51 Mersenne primes currently known.
The vertical axis plots the exponents p, which are essentially the logs base 2 of the Mersenne primes. The scale is logarithmic, so this plot suggests that the Mersenne primes grow approximately linearly on a double logarithmic scale.
Update: A new Mersenne prime was announced December 21, 2018. The graph was updated to include this new data point.