One rarely gets to write a "real" number as a tower of powers (as opposed to large numbers constructed solely for the sake of being large), so let us write the recently discovered Mersenne prime 2^74207281-1 as a short tower of 2s:

2^(74207281-very small)

2^2^26.145057410993628413

2^2^2^4.7084663330378790944

2^2^2^2^2.2352572139710255571

2^2^2^2^2^1.1604408537661861258

2^2^2^2^2^2^0.21467299217592807307

Or (tetrate 2 5.21467299217592807307) using tetration extended to real numbers by "linear" approximation.

Or, as a tower of 10s:

10^22338617.477665834124

10^10^7.3490562914116265715

10^10^10^0.8662315739486796570

(tetrate 10 2.8662315739486796570)

Each of those decimal numbers could be computed to millions of digits. There's a Procrustean feeling to this: Mersenne numbers are inherently related to powers of 2, but we are forcing this one to be a power of 10.

Given a number N, solve for X in N = (tetrate X X). Analogous problem for self-exponentiation.

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