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I've long wondered:

Is 2^(2^(2^(2^(2^(...(2)...)-1)-1)-1)-1)-1 prime for all levels of recursion?

Those are the Catalan-Mersenne numbers, and it is unknown whether c_5 is prime: http://oeis.org/A007013

May be. Each value in the series is, in binary, all 1s with the quantity of digits equal to the previous value in the series. Someone took a stab at this theory: http://www.mail-archive.com/mersenne@base.com/msg06260.html