By Stephen P. Richards

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In this chapter we wish to consider all the known prime numbers in increasing order 2, 3, 5, 7, 11, 13, ... onward and ever upward. Is there any pattern or regularity, for example, in their appearance? We must first recognize that calculating all the primes, without missing one, is a far more arduous task that just looking for a particular large but specific prime number. As a result we shall not be concerned here with prime number data out to thousands of digits, as we were with the Mercennes.

46 A NUMBER FOR YOUR THOUGHTS TABLE 3 The probability p (n) that a randomly chosen house number in a street with n houses consecutively numbered from 1 to n should have a first digit 1. n p{n) n p{n) n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 2/10 3/11 4/12 5/13 6/14 7/15 8/16 9/17 10/18 11/19 11/20 21 22 23 11/21 11/22 11/23 154 66/154 50 51 52 11/50 11/51 11/52 197 198 199 200 201 202 109/197 110/198 111/199 111/200 111/201 111/202 97 98 99 100 101 102 103 11/97 11/98 11/99 12/100 13/101 14/102 15/103 997 998 999 1000 1001 1002 1003 111/997 111/998 111/999 112/1000 113/1001 114/1002 115/1003 150 151 152 153 62/150 63/151 64/152 65/153 1999 2000 2001 1111/1999 1111/2000 1111/2001 p{n) Consider, for example, the street numbers which I laboriously extracted from Who's Who in America.

As an example, in the four horse race above where we group by eights (that is in which n =8), the value of m is 3 so that we can never expect to get more than two consecutive primes in anyone of the four rows. In this race it is obvious that bursts of speed are not sustained at all, so that overtaking the leader is likely to be a slow and tedious process if one falls too far behind. In the two-horse race with n =6, on the other hdnd, the situation is not quite so bad since the smallest prime not dividing 6 exactly is m = 5.

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