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BOX_M̃ (also stylized BOX_M~) is a large number coined by Marco Ripà.[1] [2] He claimed it to be the largest named number at the time (January 2012), although the actual winner has been Rayo's number since 2007. The number is an example of a salad number.

  • \(n\$ = {}^{n!}(n!)\) (Pickover's superfactorial)
  • \(n\widetilde{¥} = ({}^{n\$}(n\$)) \uparrow \cdots \uparrow ({}^{2\$}(2\$)) \uparrow ({}^{1\$}(1\$))\), using arrow notation.
  • \(n£ = ({}^{n\widetilde{¥}}(n\widetilde{¥})) \uparrow \cdots \uparrow ({}^{2\widetilde{¥}}(2\widetilde{¥})) \uparrow ({}^{1\widetilde{¥}}(1\widetilde{¥}))\)
  • Set \(n = G£\), where \(G\) is Graham's number.
  • \(A_1 = n£=G££\), \(A_{k + 1} = {}^{(A_k)}(A_k)\)
  • \(M_1(a) = a \uparrow^{a} a\), \(M_{k + 1}(a) = a \uparrow^{M_k(a)} a\) (in BEAF, \(M_k(a) = a \{\{1\}\} (k + 1)\))
  • \(k_1 = M_{n£}(A_{n£})!\), \(k_{i + 1} = n \uparrow^{k_i} n\)
  • \(\widetilde{R} = k_{k_{._{._{._{G£}}}}}\), where \(G\) is Graham's number, and with \(G£\) copies of \(k\). The author also coined the word "ripation" for the name of the hyperoperator \(\uparrow^{\widetilde{R}}\).
  • \(\widetilde{M}_1 = (G£ \uparrow^{\widetilde{R}} G£) \rightarrow (G£ \uparrow^{\widetilde{R}} G£) \rightarrow \cdots \rightarrow (G£ \uparrow^{\widetilde{R}} G£) \rightarrow (G£ \uparrow^{\widetilde{R}} G£)\), with \(G£ \uparrow^{\widetilde{R}} G£\) horizontal arrows, using chained arrow notation
  • \(\widetilde{M}_{k + 1} = \widetilde{M}_k \rightarrow \widetilde{M}_k \rightarrow \cdots \rightarrow \widetilde{M}_k \rightarrow \widetilde{M}_k\), with \(\widetilde{M}_k\) horizontal arrows
  • \(BOX\_\widetilde{M} = \widetilde{M}_{\widetilde{M}_1 + 1}\)

Note that in Peter Hurford's extension to chained arrows, the \(\widetilde{M}\) sequence can be more simply defined as \(\widetilde{M}_0 = G£ \uparrow^{\widetilde{R}} G£\) and \(\widetilde{M}_{k + 1} = \widetilde{M}_k \rightarrow_2 \widetilde{M}_k\).

Additionally, the paper includes the following function that isn't actually used in the definition of BOX_M̃:

  • \(n¥ = ({}^{n!}(n\$)) \uparrow \cdots \uparrow ({}^{2!}(2\$)) \uparrow ({}^{1!}(1\$))\)

Size

The function \(g(n) = n \rightarrow_2 n = \underbrace{n \rightarrow n \rightarrow \cdots \rightarrow n \rightarrow n}_{n + 1 \text{ copies of } n}\) is comparable to Conway and Guy's \(\text{CG}(n)\), or \(f_{\omega^2}(n)\) in the fast-growing hierarchy. Therefore,\(\widetilde{M}_i\) is around \(f_{\omega^2 + 1}(i)\).

Sources

  1. Ripà, Marco. The largest number ever. Retrieved February 2013.
  2. Ripà, Marco. La strana coda della serie n^n^...^n
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