A Mersenne prime number is a prime number which is one less than a power of two. About 44 have been discovered to date. For many years it was thought that all numbers of the form 2^{n} – 1 were prime. In the 16th century, however, Hudalricus Regius demonstrated that 2^{11} – 1 was 2047, with the factors 23 and 89. A number of other counter-examples were shown in the next few years. In the mid-17th century, a French monk, Marin Mersenne published a book, the Cogitata Physica-Mathematica. In that book, he stated that 2^{n} – 1 was prime for an n value of 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257.
At the time, it was apparent there was no way he could have tested the truth of any of the higher numbers. At the same time, his peers also could not prove or disprove his assertion. In fact, it wasn’t until a century later that Euler was able to demonstrate that the first unproven number on Mersenne’s list, 2^{31} – 1, was in fact prime. A century later, in the mid-19th century, it was shown that 2^{127} – 1 was also prime. Not long after that it was shown that 2^{61} – 1 was also prime, showing that Mersenne had missed at least one number in his list. In the early-20th century two more numbers were added that he had missed, 2^{89} – 1 and 2^{107} – 1. With the advent of computers checking whether numbers were prime or not became much easier, and by 1947 the entire range of Mersenne’s original Mersenne prime numbers had been checked. The final list added 61, 89, and 107 to his list, and it turned out that 257 was not in fact prime.
Nonetheless, for his important work in laying out a groundwork for later mathematicians to work from, his name was given to that set of numbers. When a number of 2^{n} – 1 is in fact prime, it is said to be one of the Mersenne prime numbers.
A Mersenne prime number also has a relationship to what are known as perfect numbers. Perfect numbers have had an important place in number-based mysticism for thousands of years. A perfect number is a number n which is equal to the sum of its divisors, excluding itself. For example, the number 6 is a perfect number, because it has the divisors 1, 2, and 3, and 1+2+3 is also equal to 6. The next perfect number is 28, with the divisors 1, 2, 4, 7, and 14. The next jumps up to 496, and the next is 8128. Each perfect number has the form 2^{n-1}(2^{n} – 1), where 2^{n} – 1 is also a Mersenne prime number. This means that in finding a new Mersenne prime number, we also focus in on finding new perfect numbers.
Like many numbers of this sort, finding a new Mersenne prime number gets more difficult as we progress, because the numbers get substantially more complex, and require much more computing power to check. For example, while the tenth Mersenne prime number, 89, can be checked quickly on a home computer, the twentieth, 4423, will tax a home computer, and the thirtieth, 132049 requires a large amount of computing power. The fortieth known Mersenne prime number, 20996011 contains more than six-million individual digits.
The search for a new Mersenne prime number continues, as they play an important role in a number of conjectures and problems. Perhaps the oldest and most interesting question is whether there is an odd perfect number. If such a thing existed, it would have to be divisible by at least eight prime numbers, and would have at least seventy-five prime factors. One of its prime divisors would be larger than 10^{20}, so it would be a truly monumental number. As computing power continues to increase, however, each new Mersenne prime number will become a bit less difficult, and perhaps these ancient problems will eventually be solved.