查看“︁雙重指數函數”︁的源代码

[[Image:Double Exponential Function.svg|right|thumb|320px|雙重指數函數（紅色）和一般實數指數冪（藍色）的比較]]

'''雙重指數函數'''是指公式為<math>f(x) = a^{b^x}</math>的[[函數]]，是指數為另一個指數冪的指數[[冪]]，在x<0時，雙重指數函數接近1，但當x>0時，雙重指數函數成長速率比指數函數還要快。

例如''a'' = ''b'' = 10時：
*''f''(−1) ≈ 1.26
*''f''(0) = 10
*''f''(1) = 10<sup>10</sup>
*''f''(2) = 10<sup>100</sup> = [[古高爾]]（googol）
*''f''(3) = 10<sup>1000</sup>
*''f''(100) = 10<sup>10<sup>100</sup></sup> = [[古戈爾普勒克斯]]（googolplex）

[[階乘]]的成長速度比指數函數還快，但比雙重指數函數慢很多。而[[迭代冪次]]和[[阿克曼函數]]的成長速度比雙重指數函數要快很多。<!-- See [[Big O notation]] for a comparison of the rate of growth of various functions.-->

==雙重指數數列==
以下是一些和雙重指數有關的數列：
* [[元數|n元]][[邏輯運算符]]的個數：
::<math>2^{2^n}</math>
*[[費馬數]]
::<math>F_m = 2^{2^m}+1</math>
*[[雙重梅森數]]
::<math>M_{M_p} = 2^{2^p-1}-1</math>

Aho和Sloane發現有許多[[整數數列]]的每一項是前一項的平方再加上一個整數，這類的數列常常可以用最接近雙重指數數列的整數來表示，且雙重指數數列中間的指數為2<ref name="aho"/>。若一整數數列的第''n''項和''n''的雙重指數成正比，Ionascu 及Stanica將這樣的整數數列稱為「幾乎雙重指數」（almost doubly-exponential），可以定義為雙重指數加上一常數後再取整數<ref>{{citation
 | last1 = Ionascu | first1 = E.
 | last2 = Stanica | first2 = P.
 | issue = 1
 | journal = Acta Mathematica Universitatis Comenianae
 | pages = 75–87
 | title = Effective asymptotics for some nonlinear recurrences and almost doubly-exponential sequences
 | volume = LXXIII
 | year = 2004}}.</ref>。

*[[西爾維斯特數列]]{{OEIS|id=A000058}}
::<math>s_n = \left\lfloor E^{2^{n+1}}+\frac12 \right\rfloor</math>
:其中''E'' ≈ 1.264084735305302為Vardi常數。
*質數2, 11, 1361, ... {{OEIS|id=A051254}}
::<math>a(n) = \left\lfloor A^{3^n}\right\rfloor</math>
:其中''A'' ≈ 1.306377883863為[[米尔斯常数]]。

==應用==
===演算法複雜度===
在[[計算複雜性理論]]中，有些[[演算法]]的[[時間複雜度]]是雙重指數，例如：
*在[[實閉域]]中的[[量詞消去]]。
*計算[[正则表达式]]的[[差集]]<ref>{{cite conference
 | last1 = Gruber
 | first1 = Hermann
 | last2 = Holzer
 | first2 = Markus
 | title = Finite Automata, Digraph Connectivity, and Regular Expression Size
 | pages = 39–50
 | booktitle = Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008)
 | url = http://www.hermann-gruber.com/data/icalp08.pdf
 | year = 2008
 | doi = 10.1007/978-3-540-70583-3_4
 | volume = 5126
 | ref = harv
 | access-date = 2013-01-23
 | archive-date = 2011-07-11
 | archive-url = https://web.archive.org/web/20110711163607/http://www.hermann-gruber.com/data/icalp08.pdf
 | dead-url = no
 }}</ref>。

===數論===
有些[[數論]]中的上限是雙重指數，例如有''n''個相異質數的奇[[完全數]]的上限為<ref>{{citation
 | last = Nielsen
 | first = Pace P.
 | journal = INTEGERS: the Electronic Journal of Combinatorial Number Theory
 | pages = A14
 | title = An upper bound for odd perfect numbers
 | url = http://www.integers-ejcnt.org/vol3.html
 | volume = 3
 | year = 2003
 | accessdate = 2013-01-23
 | archive-date = 2017-03-21
 | archive-url = https://web.archive.org/web/20170321054739/http://integers-ejcnt.org/vol3.html
 | dead-url = no
 }}.</ref>：
:<math>2^{4^n}</math>
<!--The maximal volume of a ''d''-lattice [[polytope]] with ''k'' ≥ 1 [[Integer points in convex polyhedra|interior lattice points]] is at most
:<math>(8d)^d\cdot15^{d\cdot2^{2d+1}}</math>
a result of Pikhurko.<ref>{{citation|last=Pikhurko
 | first=Oleg
 | title=Lattice points in lattice polytopes
 | journal=Mathematika
 | volume=48
 | year=2001
 | pages=pp. 15–24
 | arxiv=math/0008028
|bibcode = 2000math......8028P }}</ref>-->

自從Miller和Wheeler在1951年利用[[延遲存儲電子自動計算器|EDSAC]]找到79位數的質數之後．利用電腦找到的[[已知最大質數]]和年份之間的關係為雙重指數函數<ref>{{citation
 | last1 = Miller | first1 = J. C. P.
 | last2 = Wheeler | first2 = D. J.
 | doi = 10.1038/168838b0
 | journal = Nature
 | pages = 838
 | title = Large prime numbers
 | volume = 168
 | year = 1951
 | issue=4280|bibcode = 1951Natur.168..838M }}.</ref>。

==參考資料==
{{reflist|refs=
<ref name="aho">{{citation|first1=A. V.|last1=Aho|first2=N. J. A.|last2=Sloane|url=http://neilsloane.com/doc/doubly.html|title=Some doubly exponential sequences|journal=Fibonacci Quarterly|volume=11|year=1973|pages=429–437|accessdate=2013-01-22|archive-date=2021-05-06|archive-url=https://web.archive.org/web/20210506210458/http://neilsloane.com/doc/doubly.html|dead-url=no}}</ref>
}}

[[分類:基本特殊函数]]
[[分類:指数]]