ORGANIZE YOUR MIND --- THINK ABOUT NUMBERS -- Episode 2

                           THOSE INTERESTING PRIMES --- continued

   Prime numbers are like atoms. They are the building blocks of all integers. Every integer is either itself a prime or the product of primes. For example, 11 is a prime ; 12 is the product of the primes 2, 2, 3 ; 13 is a prime ; 14 is the product of the primes 2 and 7; 15 is the product of the primes 3 and 5 ; and so on. Some 2,500 years ago, Euclid gave a proof that "the supply of primes is inexhaustible. [ This has been discussed before, but bears repeating.]  
   Assume, said Euclid, that there is a finite number of primes. Then, one of them, call it P, will be the largest. Now consider the number Q, larger than P, that is equal to the product of the consecutive whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 . . . x P) + 1. From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into it ; each each division would leave a remainder of 1. If Q is not prime, it must be evenly divisible by some prime larger than P. On the other hand, if Q is prime, Q itself is a prime larger than P. Either possibility implies the existence of a prime number larger than the assumed largest prime P. This means, of course, that the concept of "the largest prime" is a fiction. But if there's no such beast, the number of primes must be infinite. 
   As of this writing, the largest known prime is a 909, 526-digit number formed by raising 2 to the 3, 021,377th power and subtracting 1. The prime was found on January 27, 1998, by the GIMPS project( Great Internet Mersenne Prime Search), in which 4,000 "primees" (prime number groupies) communicated over the Internet and pooled their computers for the hunt. Each of the 4,000 computers was assigned an interval of numbers to check. Roland Clarkson, a 19-year-old sophomore at California State University Dominguez Hills, was the lucky primee whose 200 Mhz Pentium-based home PC, after 46 days of running part time, examining the numbers in his assigned interval, proved the primality of 2 raised to the 3, 021, 377th power -- 1. 
   The hunt for large primes has come a long way since the seventeenth century, when Marin Mersenne, a Parisian monk, took time out from his monastic duties to search for primes. A number like 2 raised to the 3, 021th power ---1 that is the form 2 to the nth --1 is said to be a Mersenne number.  For a Mersenne number to be prime, n itself must be prime. Thus, since 2 raised to the 3, 021, 377th power --1 is prime, 3, 027, 377 must also be prime. But n being prime does not guarantee that the corresponding Mersenne number is prime. When n takes on the first four prime numbers, Mersenne primes are indeed generated : 

     For n = 2, 2 squared - 1 = 3
     For n =3, 2 raised to the third power -1 = 7 
     For n = 5, 2 raised to the 5th power -1 = 31
     For n = 7, 2 raised to the 7th power - 1 = 127 

   But when n is the fifth prime number, 11, the corresponding Mersenne number proves to be composite (2 raised to the 11th power -1 is 2,047, whose prime factors are 23 and 89) . In 1644, Mersenne himself claimed that when n took on the values of the sixth, seventh, and eighth prime numbers, namely, 13, 17, and 19, the corresponding Mersenne numbers, 2 raised to the 13th power - 1 (or 8,191), 2 raised to the 17th power - 1 (or 131, 071 ) and 2 raised to the 19th power - 1 (or 524,287) were primes. He was right. 

    The monk also made the bold claim that 2 raised to the 67th power -1 was prime.  The claim was not disputed for more than 250 years. Then, in 1903, Frank Nelson Cole of Columbia University delivered a talk with the unassuming title "On the Factorization of Large Numbers" at a meeting of the American Mathematical Society. Cole, who was always a man of very few words, walked to the board and, saying nothing, proceeded to chalk up the arithmetic for raising 2 to the sixty-seventh power. Then he carefully subtracted 1 [ getting the 21-digit monstrosity 147, 573, 952, 589, 676, 412, 927]. Without a word he moved over to a clear space on the board and multiplied out, by longhand : 

          193,707, 721 x 761, 838, 257, 287 

   The two calculations agreed. Mersenne's conjecture --if such it was --- vanished into the limbo of mathematical mythology. For the first time on record, an audience of the American Mathematical Society vigorously applauded the author of a paper delivered before it. Cole took his seat without having uttered a word. Nobody asked him a question.