For any natural number N, the number of primes not greater than it is of the order of the logarithm of N. It can be proved also that for any prime p, the next prime is less than 2p. There is no consistency, however; for instance, the nearby numbers 86 629 and 86 677 are both primes, and the virtually adjacent numbers 8 004 119 and 8 004 121 are both primes, called ‘paired primes’. Primes appear to be distributed generally without pattern, but the Mersenne primes provide something of a patterned subset. These develop the fact that 3 = 22 – 1, 7 = 23 – 1, 31 = 25 – 1, and 127 = 27 – 1 to suggest that 2n – 1 is a prime if n is a prime. But the prime n = 11 fails, as do many others. However, the formula holds true for an extended if not unlimited range, for four three-figure primes, for eight four-figure primes, and at least to n = 216 091 (giving a Mersenne prime with over 65 000 decimal digits); it provides one relatively economical means for the esoteric exercise of seeking ever larger prime numbers.
From the Oxford Dictionary of Units and Measures via Answers.com.