The French priest Marin Mersenne (1588 –1648) played an important role in 17th century number theory and also more general mathematics. Scholars inquisitive about mathematics or stumped by a difficult problem would often write to Mersenne who could direct them to a likely authority, if he didn’t know the answer himself. Today Mersenne’s name is mainly associated with numbers of the form 2n – 1; that is, numbers one less than a power of 2. To honour Mersenne these are called Mersenne numbers. It is clear that all such numbers are odd but more importantly some of them are prime; as is the case with 213466917 – 1.
Mersenne immediately understood
that if n is composite then 2n – 1 must also be composite. For example take n = 33, then
233 – 1 = 8,589,934,591 = 7 
1,227,133,513
is not a prime. However when
n is prime, the situation becomes less clear. Letting p = 2, 3, 5, and 7 yields the “Mersenne primes”
22 – 1 = 3, 23 – 1 = 7, 25 – 1 = 31, and 27 – 1 = 127
respectively. But if we take p = 11 as the exponent we get 211 – 1 = 2047 = 23 89. Mersenne was also fully aware that a prime p was not enough to guarantee 2p – 1 to also be prime. In fact he made the following assertion in his book Cognitata Physica-Mathematica:
“The only primes between 2 and 257 for which 2p – 1 is prime
are p = 2, 3, 5, 7, 9, 11, 13, 17, 19, 31, 67, 127 and 257.”
Unfortunately Mersenne missed the fact that the numbers 261 – 1, 289 – 1 and 2107 – 1 are in fact prime. But conversely 267 – 1 and 2257 – 1 turned out not to be prime at all. We can forgive Mersenne for these errors as he lived in the pre-computer age. The case of 267 – 1 was proved by Edouard Lucas (1842 – 1891) in 1876 who used an argument which did not explicitly yield any of the factors. It was not until the following century that Frank Nelson Cole did find its factors.