By Stephen P. Richards
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Additional resources for A Number for Your Thoughts
This is all very well, but what about the other sets of numbers which obey the Benford relationship; the city populations, the areas of countries, the lengths of rivers (or streets), and so forth? A similar argument does not hold for these. Why do they also obey Benford's law? This time the answer is not so certain, although some progress has been made. The first clue came from Benford's original tables themselves. Since some of his collections of numbers were of groupings involving lengths and areas, it was interesting to ask which set of units had been used for making up the tables.
In fact, with a little additional thought, it is possible to establish that the non-repeating-digit emirps must fall to zero by the tenth term, since all ten-digit non-repeating numbers must contain each of the numerals 0 through 9 once only. They must therefore all have a sum of digits equal to 1+ 2 + 3 + . + 8 + 9 = 45 which is exactly divisible by 9. Now there is a simple law of arithmetic which states that any number with a sum of digits which is divisible by 9 is itself divisible by 9. It follows that all ten-digit numbers of this kind are divisible by 9.
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. The larger 'cyclic emirp' is the six-digit number 193,939. It produces the additional cycled numbers 939,391, 393,919, 939,193, 391,939, and 919,393, emirps all! Some purists might argue that one of the five-digit cyclic forms of 11,939 (namely 19,391) is not a true emirp because of its completely reversible (that is palindromic) form, but we shall leave that point for the reader to decide upon for himself.
A Number for Your Thoughts by Stephen P. Richards