BOX_M̃ (also stylized BOX_M~) is a large number coined by Marco Ripà.[1][2][3] He claimed it to be the largest named number at the time (January 2012), although the actual winner at the time was Rayo's number. 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 \(f_{\omega^2 + 1}(i)\). This makes BOX_M̃ approximately \(f_{\omega^2 + 1}(f_{\omega^2}(\widetilde{R}))\), slightly smaller and comparable to grand thrangol.
Approximations in different notations[]
Notation | Approximation |
---|---|
Bowers' Exploding Array Function | {3,{3,3,3,{3,{3,66,1,2},2,2}},1,1,2} |
Bashicu matrix system version 2.3 = 4 |
(0)(1)(2)(2)(1)(0)(1)(2)(2)(0)(1)(2)(1)(1)(0)(1)(2)(1)[65] |
Bird's array notation | {3,{3,3,3,{3,{3,66,1,2},2,2}},1,1,2} |
DeepLineMadom's Array Notation | 3[2,1,2]3[1,1,2]3[3,2]3[2,2]66 |
Hyper-E notation | E[3]3###(E[3]3##3#65#1#2)#1#2 |
Strong array notation | s(3,s(3,s(3,s(3,66,2,2),3,2),1,1,2),2,1,2) |
Fast-growing hierarchy | \(f_{\omega^2 + 1}(f_{\omega^2}(f_{\omega+2}(f_{\omega+1}(65))))\) |
Hardy hierarchy | \(H_{\omega^{\omega^2 + 1} + \omega^{\omega^2} + \omega^{\omega+2} + \omega^{\omega+1} }(65)\) |
Slow-growing hierarchy | \(g_{\varphi(1,0,0,\varphi(\varphi(1,1,\Gamma_0),0,0))}(66)\) |
Sources[]
- ↑ Marco Ripà The largest number ever (il numero dei record) (in Italian) archived at 2012-01-14. Note that the author says the WaybackMachine returns only a partially illegible version of the original page.
- ↑ Marco Ripà Congetture su interrogativi inediti, tra speculazioni, voli pindarici e considerazioni spicciole pp. 63-66. Retrieved 2023-03-04.
- ↑ Marco Ripà La Strana Coda della serie n^n^...^n pp. 42-44. Retrieved 2023-03-04.