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Prime Numbers
Math books say that any integer that has no other divisors than itself and 1 is called a composite number. A positive prime number is any integer higher than or equal 2 that has no divisor other than itself and one. Just to remember, the number 1 is regarded as neither prime nor composite: it is a unit, and constitutes a set by itself. So the small primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, etc. There are 25 primes among the first 100 integers, but primes are irregularly distributed, and occur less frequently the farther out we go in the sequence of integers. But since Euclide´s proof, two thousand years ago, we know that the number of primes is infinite. Let´s have some patience and take a look at the proof, which is simple and elegant. Let p be a prime number, and suppose that p1, p2, p3, ..., pn, is the list of all possible primes, beginning with 2, 3, 5, ..., and out to some fixed prime pn. Now we construct the number N as follows: N=(p1 x p2 x p3 x ... x pn) + 1. As N is also an integer, it must be either a prime or a composite number, then: (a) if N is composite, then by the Fundamental Theorem of Arithmetic, it has at least two prime divisors. Now none of the given primes is a divisor of N, since there is always a remainder of 1; therefore the minimum of two prime divisors which must exist if N is composite are not in the original given list of primes. Hence there exists a new prime other than p1, p2, p3, ..., pn. (b) if N is prime, it is a new prime, being one more than than some multiple of the given primes. Thus, the assumption that p1, p2, p3, ..., pn includes all the primes is false, and the number of primes is infinite. Two other descriptions of the proof can be read in http://hermetic.nofadz.com/pns/. Well, after this "back-to-school" introduction, which drew heavily on The Basic Concepts of Elementary Mathematics, by William L. Schaaf, let´s see the largest prime number yet discovered, which was revealed last week by Michael Cameron, a 20-year-old Canadian participant in a mass computer project known as the Great Internet Mersenne Prime Search (GIMPS). The number is 2^13,466,917 - 1, and contains 4,053,946 digits. The project spent 13,000 years of computer time to find this new prime number, a feat made possible because 130,000 volunteer home users, students, schools, universities ans businesses from around the world put GIMPS on their PCs to share wasted CPU time. According to George Woltman, GIMPS founder, the finding of this prime number took two years of non-stop work. A Mersenne prime is a prime number of the form 2^p -
1, and there are now only 39 known Mersenne primes. The
35th was discovered by Joel Armengaud in 1996, the 36th
by Gordon Spence in 1997, the 37th by Roland Clarkson
in 1998, and the 38th by Nayan Hajratwala in 1999. Besides
being central to number theory, these prime numbers may
help in developing unbreakable codes and message encryptions. Related news:
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