Hey, that's cool. Surely this was already known, but not to me: naively, you expect infinitely many primes of that form. (Pretending that pi is a string of random digits and other such falsities...) The number of primes less than n is about n/ln n, so the number of primes between b^k and b^(k+1) is about (b^k / ln b) * (b/(k+1)-1/k), and the probability that a randomly-chosed number between b^k and b^(k+1) is prime simplifies to pr_k \approx (b/(k+1)-1/k) / (b-1) ln b Now we just sum this from k=0 to infinity. There's a base b constant factor of 1/(b-1) ln b, and we have to sum b/(k+1)-1/k. But of course 1/(k+1)-1/k is a telescoping sum, and the remainder is the sum of (b-1)/(k+1), which diverges. (I guess we can do this more sloppily and just observe that the sum of 1/ln(b^k) diverges.) So, to be precise, I'm claiming that if you take a random set consisting of one number with each of 1,2,3,4,... digits in base b, you expect it to contain infinitely many primes. I can't think of any reason that the numbers' digits all being initial subsequences of the same infinite sequence should seriously affect this, nor of any reason that pi should be non-generic in this context. Heh. Good luck finding the next one. --Michael Kleber kleber@brandeis.edu On Jun 23, 2004, at 9:16 AM, Robert Baillie wrote:
These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi).
The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001.
I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE.
Has anyone checked beyond this limit?
Bob Baillie
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