A double exponential function is a constant raised to the power of an exponential function. The general formula is , which grows even faster than an exponential function. For example, if a = b = 10: In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... The exponential function is one of the most important functions in mathematics. ...

• f(−1) ≈ 1.26
• f(0) = 10
• f(1) = 10¹⁰
• f(2) = 10¹⁰⁰ = googol
• f(3) = 10¹⁰⁰⁰
• f(100) = 10^{10^100} = googolplex

Factorials grow faster than exponential functions, but much slower than double-exponential functions. Compare the hyper-exponential function, which grows even faster. A googol is the large number 10100, that is, the digit 1 followed by one hundred zeros (in decimal representation). ... Googolplex is the number . ... The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a number n is the product of all positive integers less than or equal to n. ... Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

Doubly exponential sequences

The nth value of each of the following integer sequences is proportional to a double exponential function of n. Although the sequences themselves are not double exponential functions, they can be defined using formulas in which such functions appear. In mathematics, an integer sequence is a sequence (i. ...

• Fermat numbers

• Double Mersenne numbers

• Elements of Sylvester's sequence (sequence A000058 in OEIS)

where E ≈ 1.26408 is Vardi's constant (sequence A076393 in OEIS).

Aho and Sloane (1973) investigate several other sequences that, like Sylvester's sequence, can be formed by rounding to the nearest integer the values of a doubly exponential function, and provide general conditions on the values of a sequence that cause it to be of this type. In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... In mathematics, a double Mersenne number is a Mersenne number of the form where n is a positive integer. ... Sylvesters sequence consists of the coprime denominators of an Egyptian fraction that adds up to 1. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Applications

In computational complexity theory, some algorithms take double-exponential time: As a branch of the theory of computation in computer science, computational complexity theory describes the scalability of algorithms, and the inherent difficulty in providing scalable algorithms for specific computational problems. ...

• Finding a complete set of associative-commutative unifiers [1]
• Satisfying CTL⁺ (which is, in fact, 2-EXPTIME-complete) [2]

Likewise, some number theoretical upper bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most In computational complexity theory, the complexity class EXPTIME (sometimes called EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(2p(n)) time, where p(n) is a polynomial function of n. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...

a result of Nielsen (2003).[3]

See also

• Ackermann function
• Big O notation

In recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. ... Big O notation or Big Oh notation, and also Landau notation or asymptotic notation, is a mathematical notation used to describe the asymptotic behavior of functions. ...

Further reading

A. V. Aho and N. J. A. Sloane, "Some doubly exponential sequences", Fibonacci Quarterly Vol. 11 (1973), pp. 429–437.