The note Cantor's Diagonal Proof discussed the application of Cantor's diagonal argument to the list of rational numbers 1 1/2 2/1 1/3 3/1 1/4 2/3 3/2 4/1 ... where we take k=1,2,3,... and for each value of k list all the fractions n/d with n + d = k and gcd(n,d)=1, in increasing order of n. Taking the decimal representations of these numbers and substituting a new digit for each number along the diagonal we start out with [3]. 0 0 0 0 0 0 0 0 0... 0 .[1]0 0 0 0 0 0 0 0... 2 . 0[4]0 0 0 0 0 0 0... 0 . 3 3[1]3 3 3 3 3 3... 3 . 0 0 0[5]0 0 0 0 0... 0 . 2 5 0 0[9]0 0 0 0... 0 . 6 6 6 6 6[2]6 6 6... 1 . 5 0 0 0 0 0[6]0 0... 4 . 0 0 0 0 0 0 0[5]0... etc. At what point we would have to deviate from the digits of pi as we continue down this list? It turns out that we can continue to select the digits of pi until reaching the fraction 4/13, which is the 83rd fraction on the list. The 82nd digit (past the decimal point) of 4/13 = 0.307692 307692 307692 ... is a 6, which also happens to be the 82nd digit past the decimal point of the number pi. Here is the sequence of digits for pi and the corresponding sequence of digits of the rational numbers: PI: 3 1415926535 8979323846 2643383279 5028841971 6939937510 Q: 1 5030060000 6003005060 0700001530 0204300600 0500380405 PI: 5820974944 5923078164 0628620899 86... Q: 6003006750 6018301000 6947503003 46... It seems a little surprising that it takes this long to reach a pair of matching digits. Assuming the digit strings are uncorrelated and that the digits of pi are "normal", I would have expected this to be like a sequence of Poisson trials with a probability of 1/10 for each trial. The distribution of waiting times for the 1st "success" in a Poisson process with rate 1/10 is given by the gamma distribution f(t) = (1/10) e^(t/10) which, not surprisingly, has an expected value of 10. This means the expected "waiting time" for the first match is 10 digits. The probability of the first match occurring at or above the 82nd digit is e^(-82/10) = 1/3641. I suppose this unusually long waiting time is due mainly to the fact that the first "0" in pi doesn't appear until the 32nd digit, whereas the rational sequence Q has an excess of 0's, especially in the early numbers. Still, even after the 0's start to appear in pi at a more "normal" rate, they continue to miss the 0's in Q for quite a while, and in fact the first match is a 6, rather than a 0. What is known about the asymptotic distribution of digits in the Q sequence? Do they approach a "normal" distribution?

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