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Since the ↑ operation is not associative, i.e. (x↑y)↑z ≠ x↑(y↑z), this begs the question whether Fuga(3) means (3↑3)↑3 = 19683 or 3↑(3↑3) = 7625597484987. Probably the latter, since the goal is to get big numbers. Otherwise the name could be «Megafuga»
—Stephan Houben

Megafuga- is a prefix used on a number n to indicate $$^nn$$ using tetration (i.e. n pentated to 2) for the purpose of discussing how kids make up names for large numbers.[1][2]

This prefix is a retronym for fuga- in response to the non-power associativity of exponentiation. Fuga- stands for $$\overbrace{n ↑ n ↑ n ↑ ... ↑ n}^n$$, and is conjured for the purpose of how a kindergartener or primary grader can make up a number after being taught about powers. It was suggested by Stephen Houben, who asked, when hearing about Alistair Cockburn's fuga- prefix:

Since the ^ operation is not associative, i.e. $$\left(x^y\right)^z \not= x^{\left(y^z\right)}$$, this begs the question whether Fuga(3) means $$\left(3^3\right)^3 = 19,683$$ or $$3^{\left(3^3\right)} = 7,625,597,484,987$$. Probably the latter, since the goal is to get big numbers....

Alistair Cockburn has kept fuga- as the former, and named the latter "megafuga-". He has edited this on his blog.

In Hyper-E notation, this is equal to E[n]1#n, and in Poly-cell notation, this is [n][n]<1>.

The first five values of megafuga-x are 1, 4, 7,625,597,484,987, 108.0723*10153, and 10101.3357*102,184. Houben noted, using a computer, that megafuga(4) is about $$4^{1.34 \times 10^{154}}$$, somewhat larger than $$4^{10^{100}}$$, and concluded that "computing all [the] digits of megafuga(4) will never happen." The decimal expansion of that number begins and ends with: 23,610,226,714,597,313,206............36,860,456,095,261,392,896 However, the number is far too large to write out all of the digits.

For megafuga-n:

Notation Approximation
Arrow notation $$n\uparrow\uparrow n$$
Bowers' Exploding Array Function $$\{n,n,2\}$$
Bird's array notation $$\{n,n,2\}$$
Chained arrow notation $$n\rightarrow n\rightarrow 2$$
Hyperfactorial array notation $$n!1$$ lower bound $$n+1!$$ upper bound
Steinhaus-Moser Notation $$n-1[4]$$ lower bound $$n[4]$$ upper bound
Strong array notation s(n,n,2)
Fast-growing hierarchy $$f_3(n)$$
Hardy hierarchy $$H_{\omega^3}(n)$$

## See also

Googological affixes

Suffixes: -teen · -ty · -plex · -illion · -yllion · -exian · -chime · -toll · -gong · -bong · -throng · -illiob
Prefixes: gar- · fz- · fuga- · megafuga- · booga- · trooga- · googo- · googolple-
SI prefixes: deca- · hecto- · kilo- · mega- · giga- · tera- · peta- · exa- · zetta- · yotta- · ronna- · quetta-
Non-SI prefixes: 1.001 · 1.01 · 1.1 · 1.5 · 2 · 3 · 666 · 104 · 105 · 107 · 108 · 1010 · 1011 · 1013 · 1014 · 1016 · 1017 · 1019 · 1020 · 1022 · 1023 · 1025 · 1027 · 1028 · 1029 · 1030 · 1031 · 1032 · 1033 · 1034 · 1035 · 1036 · 1039 · 1042 · 1045 · 1048 · 1050 · 1051 · 1054 · 1057 · 1060 · 1063 · 1066 · 1069 · 1072 · 1075 · 1078 · 1080 · 1081 · 1084 · 1087 · 1090 · 10100 · 10120 · 10150 · 10180 · 10210 · 10240 · 10270 · 10300 · 10600 · 10900 · 101200 · 101500 · 101800 · 102100 · 102400 · 102700 · 103000 and higher

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