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Concept
Double exponential function
Summary
A double exponential function is a constant raised to the power of an exponential function. The general formula is (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10:
f(x) = 1010x
f(0) = 10
f(1) = 1010
f(2) = 10100 = googol
f(3) = 101000
f(100) = 1010100 = googolplex.
Factorials grow faster than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions.
The inverse of the double exponential function is the double logarithm log(log(x)).
A sequence of positive integers (or real numbers) is said to have doubly exponential rate of growth if the function giving the nth term of the sequence is bounded above and below by doubly exponential functions of n.
Examples include
The Fermat numbers
The harmonic primes: The primes p, in which the sequence 1/2 + 1/3 + 1/5 + 1/7 + ⋯ + 1/p exceeds 0, 1, 2, 3, ...The first few numbers, starting with 0, are 2, 5, 277, 5195977, ...
The Double Mersenne numbers
The elements of Sylvester's sequence where E ≈ 1.264084735305302 is Vardi's constant .
The number of k-ary Boolean functions:
The prime numbers 2, 11, 1361, ... where A ≈ 1.306377883863 is Mills' constant.
Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term. They show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function with middle exponent 2.
Ionaşcu and Stănică describe some more general sufficient conditions for a sequence to be the floor of a doubly exponential sequence plus a constant.
In computational complexity theory, 2-EXPTIME is the class of decision problems solvable in doubly exponential time. It is equivalent to AEXPSPACE, the set of decision problems solvable by an alternating Turing machine in exponential space, and is a superset of EXPSPACE.
Official source
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