In 1876 the French mathematician Eduard Lucas found a way to verify if some Mersenne numbers are pr

The Martin Mersenne (Mersen) (1588-1648) was a French monk who's interests were not limited to religious matters! He loved music and was the first to develop a comprehensive theory of harmony. The Mersen was he who published many of his claims because of frequent Fermat correspondence. Generally the Mersen has been one of the traffickers of mathematical ideas. The Mersen showed great interest in their primes (first is a number divisible only by the unit itself, the first are the "building block" of other numbers, but the analysis can not be done within a parenthesis for stop it here) and trying to find a formula that produces panic room a mechanism created to generate large prime concerns us even today .... The Mersenne panic room had the idea to create simply multiplying the first two several times himself and removing the unit (2 N -1) for example 2 2 2-1 = 7 In this reasoning we see many similarities with the proof Euclid for the inexperience of primes, ie got the number is divisible by many numbers and shifted by one unit in the hope that the new number will not be divisible by anyone. Quickly realized that in order to have any chance this guy should and N to be first. panic room But the reverse is not true that if N is prime and does not mean that the 2 N -1 must necessarily first. For example for N = 11 (first) have 2 N -1 = 2047 which is the first instance since divide by 23 (2047 23 = 89). The Mersenne had planned for what values of the N 2 N -1 will be first: 2, 3, 5, 7, 13, 19, 31, 67, 127, 257 2257 The number -1 is so great that hardly anyone would could challenge the forecast, so the Mersenne felt a safety when he made this version (we'll see if he did well ......).
In 1876 the French mathematician Eduard Lucas found a way to verify if some Mersenne numbers are prime. Using the method panic room demonstrated that the Mersenne was omitted from the list of numbers N, for which the 2 N -1 is prime, the 61, 89 and 107 and had mistakenly included the 67 It is worth noting, to understand how important was his method, I verified that it is indeed the first 2127 -1 with a number no less than 39 places !!! However, the 2257 -1 remained panic room outside the capabilities of Lukas. And I will stay out of the possibilities of the world in general to come Lemer 1930 (at the age of 25 years please) to improve the method of Likas .The proof is simple to implement, but a mystery even today the arrest. So Lemer, reversing panic room the problem, panic room showed that the 2 N -1 is prime only if divides panic room another number called Likas-Lemer and denoted by LN .The creation of this number could be said to resemble the Fibonacci sequence (an = an -1 + an -2), namely the creation of the following results from the previous ones. So to find the LN we raise in the previous block and remove panic room two words: LN = (LN -1) 2 -2 For example for N = 3 we starting L 3 = 14. So we have L 4 = 194, L 5 = 37634, L 6 = 1416317955, ... .. For example, two 5 -1 = 31 divides the L 5 = 37634 so the number 2 of 5 Mersen -1einai first. With this simple test "dropped" the last castle was raise the Mersen and have proven that 2257 -1 is not first ...! From what seemed the list of Mersen was empirical and this little tarnished reputation. Although they had checked all of the first estimates of Mersen the "hunting" of large numbers Mersen had just begun. The Lemer 1952, with the help of a computer built for this purpose, exceeded the computational capabilities of the human mind finding the 2607 -1 first. Within a year Raphael Robinson had already kataripsei other three times the record, and the largest known was now the first two 2,281 -1 !!! So entering the computer age, until the mid-1990s, large companies began to exploit panic room the ever increasing power of computational systems for the production of large prime Mersen. A typical example is when Paul and David Gage Slovinski with the help of computer Cray, and shoot down one record after another, announced in 1996 the seventh to the first 2 1257787 -1, a number with 378 632 digits. Today in search of his first major Mersen dominates who else from the Internet (Internet) known as the Great Internet Mersenne Prime Search (Great online survey for the first Mersen) or simply GIMPS. The initiator of this Gouoltman enlists computers panic room around the world and exploit the computing power by putting panic room them to work