Distribution of Digits in Integers

If you look at a long run of the decimal digits of an irrational number like e, iT or V/2, you will probably get the impression that each digit occurs about as often as any other digit, so that about one-tenth of the digits are (say) 2's. Except for numbers deliberately constructed to have lop-sided distributions of digits, you would find the same phenomenon for pairs of digits (which are single digits in base 100) and generally for digits in any base. To make statements like these precise, we need a little terminology. Given a number N, written in base b, let there be dn occurrences of the digit d among the first n digits of N. Then if dnIn -> 1/b as n -> oo, and if this holds for every base b and every digit we mrght select, we call the number N normal. As a matter of fact, nobody has ever proved that e, IT or V2 is normal, but we do know that there are lots of normal numbers. More precisely, we can find a sequence of intervals, of total length as small as we like, in which all the non-normal numbers lie; hence there must be many more normal numbers than non-normal ones. In the terminology of measure theory, the set of non-normal numbers has measure zero. This situation is usually described briefly by saying that almost all real numbers are normal. For a proof see, for example, [2] or [4]. The fact that most numbers are normal has some startling consequences. We can translate English text into a numerical code by assigning numbers to the letters of the alphabet and the punctuation marks, for example by using the ASCII code, which can be implemented on many computers. Suppose then that we translate the whole Encyclopaedia Britannica into numerical code and read the result as a single (rather large) integer B. This is a single digit in base B + 1 (or any larger base), and so for almost every real number (all the normal ones) the digit B occurs infinitely often in the representation of that number in any base greater than B. If we take the base to be a power of 10, saying that the "digit" B occurs infinitely often in the expansion of a given number to that base is equivalent to saying that the numerical representation of the Encyclopaedia occurs infinitely often in the ordinary decimal expansion of the same number. We cannot, of course, observe even a single occurrence, since the length of the decimals required to show it would be unreasonably large. Notice, by the way, that we do not know that the numerical representation of the Encyclopaedia occurs in the decimal expansion of e, I, V/2, or similar numbers, since none of the standard numbers that arise in elementary mathematics is known to be normal. Explicit examples of normal numbers are not easy to find. One is the number 0.12345678910111213. obtained by writing the consecutive integers, in base 10, in a row and putting a decimal point in front of it ([4], p. 112). We cannot expect the decimal representations of integers to be normal, since each one has only finitely many non-zero digits. However, if you look at some large integers (factorials, for example) written out in base 10, you will quite likely again get the impression that the digits are fairly uniformly distributed. One might guess that this should be the case for most integers, with some interpretation of "most". Now it is known (see, for example, [5]) that the integers whose decimal representations contain no zeros are rare enough so that 11/n, summed over these integers, converges; whereas we learn in elementary calculus that 11/n, summed over all integers, diverges. This suggests a way of