What is Mersenne's number?

Primening Mersenne is a prime number that is one less than the power of two. About 44 have been discovered so far. For many years it was assumed that all numbers of form 2 n - 1 were first -class. In the 16th century, however, Hudalricus Regius showed that 2

11

-1 was 2047, with factors 23 and 89. In the next few years several other opposites were shown. In the mid-17th century, the French monk Marin Mersenne published the book The Cogitata Physica-Matematica . In this book, he said that 2 n - 1 was the main for n value 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257.

At that time it was clear that there was no way to test the truth about any of the higher numbers. At the same time, his peers could not prove or refute his claim. In fact, it was only a century later that Euler was able to prove that the first unproven number on Mersenne's list,2 31 - 1, was actually first -class. A century later, in the middle of the 19th century, it was shown that 2

127

-1 was also first-class. Shortly thereafter, it turned out that 2 61 - 1 was also first -class, which shows that Mersenne missed at least one number in his list. At the beginning of the 20th century, two more numbers were added, which was missing, 2

89 -1 and 2

107 -1. With the advent of computers that checked whether the numbers were first -class or not much easier, and by 1947 a number of original Mersenne's primary numbers were checked. The final list added to his list 61, 89 and 107, and it turned out that 257 was not really prime.

However, for his important work on the distribution of the foundation for the later mathematics from which it was possible to work, his name was given by this set of numbers. In fact, when it is 2 ^{n - 1, it is said that it is one of the prime numbers of Mersenne.}

and Japrimed Number Rsenne also has a relation to thatwhat is called perfect numbers. Perfect numbers had an important place in the mystics based on the number for thousands of years. The perfect number is the number n , which equals the sum of its divisors, with the exception of itself. For example, number 6 is a perfect number because it has a divisor of 1, 2 and 3 and 1+2+3 equals 6. Another perfect number is 28, with divisors 1, 2, 4, 7 and 14. Other jumps up to 496 and another is 8128. 2 This means that in finding a new prime mersenne we also focus on finding new perfect numbers.

As well as many numbers of this kind, finding a new prime Mersenne is more difficult, as we proceed because the numbers are much more complex and require much more computing forces to control. For example, while the tenth Mersenne Prime Number, 89, can be quickly checked on home computer, twenty, 4423, will tax home computer and thirtieth, 132049 requires a large number of computing power. Forty -Focked Prime Minister Mersenne, 20996011, contains more than six million individual numbers.

The search for a new prime number Mersenne continues because they play an important role in a number of conjecture and problems. Perhaps the oldest and most interesting question is whether there is a special perfect number. If such a thing existed, it would have to be divisible at least eight first -class numbers and would have at least seventy -five first -class factors. One of his main divisors would be greater than 10 20

, so it would be a truly monumental number. However, as the computational force continues to increase, but every new Mersenne number becomes slightly less difficult, and perhaps these ancient problems will eventually be solved.