A double exponential function is a constant

Constant (mathematics)

In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable In mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition...

 raised to the power of an exponential function

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

. The general formula is , which grows much more quickly than an exponential function. For example, if a = b = 10:

• f(−1) ≈ 1.26
• f(0) = 10
• f(1) = 10¹⁰
• f(2) = 10¹⁰⁰ = googol
  Googol
  A googol is the large number 10100, that is, the digit 1 followed by 100 zeros:The term was coined in 1938 by 9-year-old Milton Sirotta , nephew of American mathematician Edward Kasner...
  
  
• f(3) = 10¹⁰⁰⁰
• f(100) = 10¹⁰¹⁰⁰ = googolplex.

Factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

s grow faster than exponential functions, but much slower than double-exponential functions. The hyper-exponential function

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

 and Ackermann function

Ackermann function

In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive...

 grow even faster. See Big O notation

Big O notation

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

 for a comparison of the rate of growth of various functions.

The inverse of a double exponential function is a double logarithm.

Doubly exponential sequences

Aho and Sloane observed that in several important integer sequence

Integer sequence

In mathematics, an integer sequence is a sequence of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms...

s, each term is a constant plus the square of the previous term, and show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two. Integer sequences with this squaring behavior 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
  Sylvester's sequence
  In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are:...
  
   

where E ≈ 1.264084735305 is Vardi's constant.

• The number of k-ary
  Arity
  In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
  
   operators
  Logical connective
  In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
  
  :

More generally, if the nth value of an integer sequences is proportional to a double exponential function of n, Ionascu and Stanica call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly-exponential sequence plus a constant. Additional sequences of this type include

• The prime numbers 2, 11, 1361, ...

where A ≈ 1.306377883863 is Mills' constant.

Algorithmic complexity

In computational complexity theory

Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...

, some algorithms take double-exponential time:

• Decision procedures for Presburger arithmetic
  Presburger arithmetic
  Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely...
  
  
• Computing a Gröbner basis
  Gröbner basis
  In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
  
  
• Finding a complete set of associative-commutative unifiers
• Satisfying CTL⁺ (which is, in fact, 2-EXPTIME
  2-EXPTIME
  In computational complexity theory, the complexity class 2-EXPTIME is the set of all decision problems solvable by a deterministic Turing machine in O time, where p is a polynomial function of n....
  
  -complete)
• Quantifier elimination
  Quantifier elimination
  Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification...
  
   on real closed field
  Real closed field
  In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
  
  s takes at least doubly-exponential time (but is not even known to be computable in ELEMENTARY)
• Calculating the complement
  Complement (set theory)
  In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
  
   of a regular expression
  Regular expression
  In computing, a regular expression provides a concise and flexible means for "matching" strings of text, such as particular characters, words, or patterns of characters. Abbreviations for "regular expression" include "regex" and "regexp"...
  
  

Number theory

Some number theoretical

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

 bounds are double exponential. Odd perfect numbers with n distinct prime factors are known to be at most
a result of Nielsen (2003).

The largest known prime number in the electronic era has grown roughly as a double exponential function of the year since Miller

J. C. P. Miller

Jeffrey Charles Percy Miller was an English mathematician and computing pioneer. He worked in number theory and on geometry, particularly polyhedra, where Miller's monster refers to the great dirhombicosidodecahedron....

 and Wheeler found a 79-digit prime on EDSAC

EDSAC

Electronic Delay Storage Automatic Calculator was an early British computer. The machine, having been inspired by John von Neumann's seminal First Draft of a Report on the EDVAC, was constructed by Maurice Wilkes and his team at the University of Cambridge Mathematical Laboratory in England...

1 in 1951.

Theoretical biology

In population dynamics

Population dynamics

Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

 the growth of human population is sometimes supposed to be double exponential. Gurevich and Varfolomeyev experimentally fit


where N(y) is the population in year y in millions.

Physics

In the Toda oscillator model of self-pulsation

Self-pulsation

Self-pulsation takes place at the beginning of laser action.As the pump is switched on, the gainin the active medium rises and exceeds the steady-state value...

, the logarithm of amplitude varies exponentially with time (for large amplitudes), thus the amplitude varies as double-exponential function of time.