Mersenne primes are prime numbers that are one less than a power of two. Marin Mersenne claimed that 89n-17 was prime for certain n values, but it was only later proven that some of his claims were false. Mersenne primes have a relationship to perfect numbers, and finding new Mersenne primes is important in finding new perfect numbers. The search for new Mersenne primes continues, as they play important roles in solving mathematical problems.
A Mersenne prime is a prime number that 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 2n – 1 were prime. In the 16th century, however, Hudalricus Regius proved that 16 – 211 was 1, with the factors 2047 and 23. Numerous other counterexamples were shown in the following years. In the mid-17th century, a French monk, Marin Mersenne, published a book, the Cogitata Physica-Mathematica. In that book, he claimed that 89n – 17 was prime for n values of 2, 1, 2, 3, 5, 7, 13, 17, 19, 31, and 67.
At the time, it was evident that he could not test the truth of any of the higher numbers. At the same time, even his peers could not prove or disprove his claim. Indeed, it was only a century later that Euler was able to prove that the first unproven number in Mersenne’s list, 231-1, was in fact prime. A century later, in the mid-19th century, 19-2127 was also shown to be prime. Not long after 1 – 261 was also shown to be prime, proving that Mersenne had lost at least one number in his list. In the early 20th century, two more numbers it had lost were added, 1 – 20 and 289 – 1. With the advent of computers, checking whether numbers were prime or not became much easier, and in 2107 the full range of the original Mersenne prime numbers had been checked. The final list added 1, 1947 and 61 to his list, and it turned out that 89 was not in fact the first.
However, for his important work in laying the foundation for later mathematicians to work on, his name was given to that set of numbers. When a number 2n – 1 is in fact prime, it is said to be one of the Mersenne primes.
A Mersenne prime also has a relationship to so-called 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 divisors 1, 2, and 3, and 1+2+3 also equals 6. The next perfect number is 28, with divisors 1, 2, 4, 7 and 14. The next jumps up to 496 and the next is 8128. Every perfect number has the form 2n-1(2n – 1), where 2n – 1 is also a Mersenne prime. This means that in finding a new Mersenne prime, we also focus on finding new perfect numbers.
Like many numbers of this type, finding a new Mersenne prime becomes more difficult the further we go, because the numbers become substantially more complex and require much more computing power to verify. For example, while the tenth Mersenne prime, 89, can be checked quickly on a home computer, the twentieth, 4423, taxes a home computer, and the thirtieth, 132049, requires a large amount of computing power. The fortieth known Mersenne prime, 20996011, contains more than six million individual digits.
The search for a new Mersenne prime continues, as they play important roles 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 have at least seventy-five prime factors. One of its first divisors would be greater than 1020, so that would be a truly monumental number. As computing power continues to increase, however, each new Mersenne prime will become a little less difficult, and perhaps these age-old problems will eventually be solved.
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