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By Stephen P. Richards

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He even proposed an equation for this 'law' by noting that the distribution in Figure 3 is very closely given by P(n) = log[(n+l)/n] in which P (n) is the probability that the first digit will be n, and log stands for the logarithm to the base ten, which is the normal logarithm found in log tables. At this point it is perhaps desirable to say a word or two about what base-ten logs really are, although it is not necessary for the reader to know anything about logarithms in order to enjoy the Benford story.

Can we perhaps write down some polynomial in powers of n represent it; or maybe some other kind of function? Karl Friedrich Gauss, perhaps the greatest mathematician of his time (whom we met earlier in Chapter 2), asked himself just this question when he was but fifteen years old. 0 where we have added several more powers of 10 in n made possible by the modern-day computerization of the problem. 3 when we go from one power of ten to the next. 3 behavior as connected with what is called the natural logarithm of ten.

Before leaving the emirps we might mention two of them which are very special. In listing the emirps up to ten million, two have appeared which are completely cyclic. By this we mean that the process of moving the first digit to the rear, and repeating this operation as often as you please, generates only other emirps. The smaller of these two 'cyclic emirps' is the five-digit number 11,939. By the cycling process described above it generates the additional emirps 19,391, 93,911, 39,119, and 91,193.

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