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Pentation refers to the 5th hyperoperation starting from addition. It is equal to \(a \uparrow\uparrow\uparrow b\) in Knuth's up-arrow notation and since it is repeated tetration, it produces numbers that are much larger. Pentation can be written in array notation as \(\{a,b,3\}\), in chained arrow notation as \(a ightarrow b ightarrow 3\) and in Hyper-E notation as E(a)1#1#b. Pentation is less known than tetration, but there are a few googologisms employing it: 3 pentated to 3 is known as tritri, and 10 pentated to 100 is gaggol. Tim Urban calls pentation a "power tower feeding frenzy".

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  • Pentation
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  • Pentation refers to the 5th hyperoperation starting from addition. It is equal to \(a \uparrow\uparrow\uparrow b\) in Knuth's up-arrow notation and since it is repeated tetration, it produces numbers that are much larger. Pentation can be written in array notation as \(\{a,b,3\}\), in chained arrow notation as \(a ightarrow b ightarrow 3\) and in Hyper-E notation as E(a)1#1#b. Pentation is less known than tetration, but there are a few googologisms employing it: 3 pentated to 3 is known as tritri, and 10 pentated to 100 is gaggol. Tim Urban calls pentation a "power tower feeding frenzy".
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abstract
  • Pentation refers to the 5th hyperoperation starting from addition. It is equal to \(a \uparrow\uparrow\uparrow b\) in Knuth's up-arrow notation and since it is repeated tetration, it produces numbers that are much larger. Pentation can be written in array notation as \(\{a,b,3\}\), in chained arrow notation as \(a ightarrow b ightarrow 3\) and in Hyper-E notation as E(a)1#1#b. Pentation is less known than tetration, but there are a few googologisms employing it: 3 pentated to 3 is known as tritri, and 10 pentated to 100 is gaggol. Sunir Shah uses the notation \(a * b\) to indicate this function. Jonathan Bowers calls it "a to the b'th tower". Sbiis Saibian proposes \(_{b \leftarrow}a\) in analogy to \({^{b}a}\) for tetration, though he usually uses up-arrows. Pentational growth rate is comperable to \(f_4(n)\) in the fast-growing hierarchy. A strip from the webcomic Saturday Morning Breakfast Cereal suggested the name "penetration" in humorous analogy with sexation. Tim Urban calls pentation a "power tower feeding frenzy". In Notation Array Notation, it is written as (a{3,3}b).
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