June 3, 2011
Numbers of the form Mn = 2n−1 are known as Mersenne numbers, named for the French monk Marin Mersenne who studied them early in the seventeenth century. Of the infinite set of Mersenne numbers, 47 are currently known to be prime; see Sloane’s A000043 for a list of their indices. Mersenne primes can be identified by the Lucas-Lehmer test, devised by Édouard Lucas in the 1870s and cast into its modern form by Derrick Lehmer in 1930:
For p an odd prime, the Mersenne number 2p−1 is prime if and only if 2p−1 divides Sp−1 where Sn+1 = Sn2−2 and S1 = 4.
The special form of Mersenne primes makes them easy to identify, and for many years the largest known prime has been a Mersenne prime. A cooperative project on the internet, the Great Internet Mersenne Prime Search (GIMPS), has found all of the recent Mersenne primes, because the numbers have grown so large that a single computer can’t handle the workload.
Your task is to use the Lucas-Lehmer test to find the Mersenne primes through M256. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.