* Warren Smith <warren.wds@gmail.com> [Jan 30. 2012 17:09]:
From: David Wilson <davidwwilson@comcast.net> [...]
Now suppose p > 10^d and p has primitive root 10. Because of the primitive root 10, every n with 1 <= n < p is expressible as n = 10^k mod p for some 0 <= k < p-1. By our previous paragraph, there is then a fraction of the form (10^k mod p) / p starting with any desired d-digit block B. From this we conclude that every d-digit block occurs somewhere in the decimal expansion of 1/p.
--this is a nice theorem, and it might be a new one. It is also of some importance because it proves a randomness property of the Marsaglia type of pseudorandom number generator.
To determine if it is new, I suggest looking to see if it is in Hardy & Wright chapter 9 as step one. Frankly though, it looks too easy to have escaped previous notice.
Without really having followed through: This rings the "shift register sequences"-bell for me. So a good source about these might be worth checking. I recommend "the bible": Rudolf Lidl, Harald Niederreiter: {Finite fields}, Cambridge University Press, second edition, (1997); Almost the same (and sufficient for what we care about) is: Rudolf Lidl, Harald Niederreiter: {Introduction to finite fields and their applications}, Cambridge University Press, revised edition, (1994)
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