By Stephen P. Richards
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Extra resources for A Number for Your Thoughts
When continued to 100 terms, the upper series gives the first digit distribution 52 A NUMBER FOR YOUR THOUGHTS 30, 17, 13, 10,7,7,6,5,5 while the lower gives 29, 19, 12,8,8,7,7,5,5. These sorts of series are called geometric sequences. Could it therefore be that nature tends to count geometrically, that is by powers like 2, 4, 8,16, rather than by 1,2,3, 4? This in essence is the suggestion which Benford himself put forward when he first commented on the entire phenomenon. It implies that in a list of 'amomalous numbers', generated by some count of natural phenomena which covers a wide range, there is a tendency for the step between successive numbers (when arranged in ascending order) to be a fixed fraction of the preceding one.
We also show, in Table 4, the equivalent probabilities for finding a house number starting with the numeral 9. As a function of n it also has a sawtooth shape and is sketched in Figure 4b. 111) and 48 A NUMBER FOR YOUR THOUGHTS TABLE 4 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 9. 0111). Again it is clear that, except for streets with 9, 99, 999, 9,999, ... houses, the random chance of finding a house number with first digit 9 is not 1/9 but is always smaller than 1/9.
Needless to say, it is not known whether prime numbers of this form exist to arbitrarily large values. 42 A NUMBER FOR YOUR THOUGHTS Related to the repunit prime numbers are those which are all of this kind except for the first or last digit. Some examples of prime numbers of this form are 111,111,113 11,111,111,113 11,111,117 11,111,119 the last two making a rather remarkable pair of prime twins. Other examples, with the differing digit at the front of the number, are 61,111,111 71,111,111 31,111,111,111 Of the prime numbers which have the numerals in their 'proper' order 1, 2, 3, 4, ...