[Math Lair] Prime Gaps

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As we proceed higher into the realm of the natural numbers, the density of the prime numbers slowly thins out. This means that strings of composite numbers between primes grow ever longer. The smallest gap of composite numbers between primes is zero, between the primes 2 and 3. Between all other consecutive primes there must be at least one composite number (since all even numbers are composite). These primes are called twin primes, but it is not known whether the number of twin primes is infinite or not.

Between the numbers 1 and 100, the largest gap is between the primes 89 and 97, consisting of seven composite numbers. The largest gap less than 1000 is between 887 and 907, consisting of 19 composites. We can find a gap between prime numbers that is as large as we desire. To do this, examine the sequence (N + 1)! + 2, (N + 1)! + 3, . . ., (N + 1)! + (N + 1) (where ! is the factorial operator). The first number in this sequence is divisible by 2 (since (N + 1)! is divisible by 2 whenever N >= 1). Similarly, the second number is divisible by 3, the third by 4, and so on. Therefore, all of these numbers are composite and we have found a prime gap of length N. In practice, this procedure isn't too efficient. For example, generating a sequence of nine consecutive composites involves calculating 10! which is 3,628,800. By inspection, we can find a prime gap of 13 between 113 and 127.

The largest gap between primes that has been located by inspection is a gap of 803 composite numbers between 90,874,329,411,493 and 90,874,329,412,297, which as found in 1989 by J. Young and A. Potler.