| abstract | - BOX_M̃ (also stylized BOX_M~) is a large number coined by Marco Ripà. 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 \(\uparrow^{\widetilde{R}}\).
* \(\widetilde{M}_1 = (G£ \uparrow^{\widetilde{R}} G£) ightarrow (G£ \uparrow^{\widetilde{R}} G£) ightarrow \cdots ightarrow (G£ \uparrow^{\widetilde{R}} G£) ightarrow (G£ \uparrow^{\widetilde{R}} G£)\), with \(G£ \uparrow^{\widetilde{R}} G£\) horizontal arrows, using chained arrow notation
* \(\widetilde{M}_{k + 1} = \widetilde{M}_k ightarrow \widetilde{M}_k ightarrow \cdots ightarrow \widetilde{M}_k ightarrow \widetilde{M}_k\), 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 \(\widetilde{M}_{k + 1} = \widetilde{M}_k ightarrow_2 \widetilde{M}_k\).
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