If you want to know where the next Mersenne prime number might be:
There is a conjecture that the values of the Mersenne prime exponents follow the function 2^{\gamma x} on a statistical average and thus can be approximated by a straight line (in a logarithmic representation). The plot shows the straight in red color.
Here, \gamma is the Euler Mascheroni constant. If we use the method described in my book for determining the sequence function (which is identical with the first 53 values of the Mersenne prime number exponents, including the ‘1’), (marked with yellow color) and extrapolate the sequence function further on for a few indices (marked in blue color), then we get the following graph:
Of course, there is no (mathematical) reason at all to assume that the next Mersenne prime is really located in the near of 2^{279143407}-1 (marked in green color). However, the extrapolated ‚trend‘ of the sequence function was too beautiful and the author could not could not resist plotting the ‚extrapolated‘ Mersenne prime exponents up to !
If you look at the graph, you can’t get rid of the feeling that the next Mersenne prime exponents are pushing towards higher values again, since the last values are significantly below the predicted values.