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Is anything known about these problems?
If we make a string S of 0's and 1's with 1 in n'th position if the the nth prime $p_n$ is of the form $1+m 2^{9^{9^{9^{9}}}}$, else 0, does every finite string of 0's and 1's appear in S infinite times?

And S2: Write 0 in n'th position if $p_n - p_{n-1} < p_{n-1} - p_{n-2} - 2^{9^{9^{9^{9}}}}$, else 1. Does every finite string of 0's and 1's appear in S2 infinite times?

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I believe I may have figured out what you meant by the last question -- please check that the clarified version is what you meant. –  joriki Nov 5 '11 at 6:30

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