What are the chances that the Lucas-Lehmer test will find a new Mersenne prime number? A simple approach is to repeatedly apply the observation that the chance of finding a factor between 2

^{X} and 2

^{X+1} is about 1/x. For example, you are testing 2

^{10000139}-1 for which trial factoring has proved there are no factors less than 2

^{64}. The chance that it is prime is the chance of no 65-bit factor * chance of no 66 bit factor * ... * chance of no 5000070 bit factor. That is:

64 65 5000069

-- * -- * ... * -------

65 66 5000070

This simplifies to 64 / 5000070 or 1 in 78126. This simple approach isn't quite right. It would give a formula of how_far_factored divided by (exponent divided by 2). However, more rigorous work has shown the formula to be (how_far_factored-1) / (exponent times

Euler's constant (0.577...)). In this case, 1 in 91623. Even these

more rigourous formulas are unproven.