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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 (more recently it was surpassed by BIG FOOT). 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