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Math Trek

Closing the Gap on Twin Primes

By
2:08pm, July 15, 2005

More than 2,000 years ago, Euclid of Alexandria (325–265 B.C.) provided a simple proof that the sequence of prime numbers continues forever. A prime is a whole number (other than 1) that's evenly divisible only by itself and 1. This definition leads to the following sequence of numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, and so on.

Suppose there is a finite number of primes, Euclid argued. This means there's also a largest prime, n. Multiply all the primes together, then add 1: (2 x 3 x . . . x n) + 1. The new number is certainly bigger than the largest prime.

If the initial assumption is correct, the new number can't be a prime. Otherwise, it would be the largest. Hence, it must be a composite number and divisible by a smaller number. However, because of the way the number was constructed, any known prime, when divided into the new number, leaves a remainder of 1. Therefore, the initial assumpt

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