Pentatri is equal to {3,3,3,3,3} or {3,5 (1) 2} (3 powergigoexploded to 3) in BEAF.[1] or 5 & 3, using the array of operator. The term was coined by Jonathan Bowers.
It is also quite close to the lower bound for \( G(16) \) in the Goodstein function, which illustrates that large numbers can appear in the modern mathematics.
Chris Bird noted that Friedman's n(k) function is comparable to {3,k (1) 2}, so pentatri should be in ballpark with n(5).[2]
Etymology
The name of this number is based on the words "penta-" (greek: five) and "tri-" (greek: three), meaning that there are 5 3's in the array.
Approximations in other notations
| Notation | Approximation |
|---|---|
| BEAF | \(\{3,3,3,3,3\}\) or \(\{3,5(1)2\}\) (both equal) |
| Bird's array notation | \(\{3,3,3,3,3\}\) (exactly equal) |
| Hyper-E notation | E(3)3####3 |
| X-Sequence Hyper-Exponential Notation | \(3\{X^3\}3\) |
| Multivariable Ackermann | \(A(1,0,0,0,3)\) |
| Strong array notation | s(3,4,4,3,3) |
| Fast-growing hierarchy | \(f_{\omega^{3}}(3)\) |
| Hardy hierarchy | \(H_{\omega^{\omega^{3}}}(3)\) |
| Slow-growing hierarchy | \(g_{\varphi(2,3,3,0)}(3)\) |
Sources
- ↑ Bowers, Jonathan. Infinity Scrapers. Retrieved January 2013.
- ↑ Bird, Chris. Linear Array Notation. Retrieved 03.03.2013.
See also
Tritri series: tritri · tritet · tripent · trisept · tridecal · grand tridecal
Tetratri series: tetratri · supertet · general(plex)
Pentatri series: pentatri · superpent · pentadecal(plex)
Hexatri series: hexatri · superhex · hexadecal(plex)
Heptatri series: heptatri · supersept · heptadecal
Iteral series: superoct · octadecal · superenn · ennadecal · iteral · ultatri
Dupertri series: dupertri · duperdecal · truperdecal · quadruperdecal
Latri series: latri · emperal(plex) · hyperal(plex) · admiral
Dutritri series: dutritri · dutridecal
Dimentri series: dimentri · dulatri · trilatri · trimentri
Triakulus series: triakulus · tridecatrix
Big boowa series: big boowa · great big boowa · grand boowa
Tiaokhiao's extensions: trihex · trioct · triennet · triundecal · tridodecal · tritriplex · tritriplexian · grand tritri · tetrapent · tetrahex · grand tetratri · octatri · enneatri · decatri · undecatri · dodecatri
