Down-arrow notation
Type3-argument
Based onExponentiation
Growth rate\(f_{\omega}(n)\)

Down-arrow notation is a left-associative extension for addition, multiplication and exponentiation. It is a weaker version of arrow notation.

Formally, \(a \downarrow \cdots \downarrow b\) for positive integers \(a\) and \(b\) is defined as follows: \begin{eqnarray*} a \downarrow^n b = \left\{ \begin{array}{cl} a^b & (n = 1) \\ a & (n > 1, b = 1) \\ (a \downarrow^n (b-1)) \downarrow^{n-1} a & (n > 1, b > 1) \end{array} \right. \end{eqnarray*} Here, \(a \downarrow^n b\) for a positive integer \(n\) is a shorthand for \(a \downarrow\cdots\downarrow b\) with n down-arrows.

When \(n = 2\), \(a \downarrow\downarrow b = a^{a^{b-1}}\). The inequality \(a \downarrow\downarrow a = a^{a^{a-1}} < a^{a^a} = a \uparrow\uparrow 3\) is useful when bounding down-arrows in terms of up-arrows.

It can be shown that \(a \downarrow^{2n-1} b \ge a \uparrow^n b\) for \(a, b, n \ge 1\).

Down-arrow notation is not as important in googology as the up-arrow notation, but it is used in the definition of Clarkkkkson.

Examples

  • \(3 \downarrow\downarrow 3 = ((3 \downarrow\downarrow 1)\downarrow 3)\downarrow 3 = (3^3)^3 = 3^9 = 19,683\)
  • \(3 \downarrow\downarrow\downarrow 2 = (3 \downarrow\downarrow\downarrow 1)\downarrow\downarrow 3 = 3\downarrow\downarrow 3\)
  • \(3 \downarrow\downarrow\downarrow 3 = (3 \downarrow\downarrow\downarrow 2)\downarrow\downarrow 3 = 19,683 \downarrow\downarrow 3 = 19,683^{19,683^2} = 3^{3^{20}}\)

Using BCalc, we find \(3 \downarrow\downarrow\downarrow 3 \approx 2.198197017816742204069545026183 \cdot 10^{1,663,618,948}\)

Bounds

  • \(a\downarrow\downarrow\downarrow b\approx a\uparrow\uparrow(b+1)\)
  • \(a\downarrow^4 b\approx a\uparrow\uparrow(a(b-1)+1)\)
  • \(a\downarrow^5 b\approx(a\uparrow\uparrow)^{b-1}(a(a-1)+1))>a\uparrow\uparrow\uparrow b\)
  • \(a\downarrow^6 b\approx(a\uparrow\uparrow)^{(a-1)(b-1)}(a(a-1)+1)>a\uparrow\uparrow\uparrow((a-1)(b-1)+1)\)

See also

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