Tripent visualized using iterated power towers.
"Fashion" view of tripent using iterated power towers with arrows.
The tripent is equal to \(\lbrace5,5,5\rbrace\) or \(5\lbrace5\rbrace5\) in BEAF, \(5\uparrow\uparrow\uparrow\uparrow\uparrow5\) in arrow notation, or 5 heptated to 5.[1]. The term was coined by Jonathan Bowers. This is the fifth Ackermann number.
Googology Wiki user Hyp cos calls this number a trienpent, and it's equal to s(5,5,5), s(5,2,6), s(5,5,1,2), s(5,1,1,3), or s(5,1,1,1,2) in strong array notation. [2]
It can be computed during the following process:
- \(A(1)\) = 5
- \(A(2)\) = 5^5^5^5^5 (A(1) 5's)
- \(A(3)\) = 5^5^5^...5^5^5, with A(2) 5's.
- \(A(4)\) = 5^5^5^5^5^......5^5^5^5, with A(3) 5's
- Continue, let \(A(n)\) be 5^5^5^5^......^5^5^5 with \(A(n-1)\) 5's
- Tripent is . \(A^{A^{A^{A^{5}(1)}(1)}(1)}(1)\)
Etymology
The name of this number is based on tri- (three) and penta- (five), meaning that there are 3 5's in the array.
Approximations
| Notation | Approximation |
|---|---|
| Arrow notation | \(5\uparrow\uparrow\uparrow\uparrow\uparrow5\) (exact) |
| Bowers' Exploding Array Function | \(\{5,5,5\}\) (exact) |
| Chained arrow notation | \(5\rightarrow5\rightarrow5\) (exact) |
| Hyper-E notation | \(E(5)1\#1\#1\#1\#5\) (exact) |
| Hyperfactorial array notation | \(6!4\) |
| Fast-growing hierarchy | \(f_6(4)\) |
| Hardy hierarchy | \(H_{\omega^6}(4)\) |
| Slow-growing hierarchy | \(g_{\varphi(4,0)}(5)\) |
Sources
- ↑ Bowers, Jonathan. Infinity Scrapers. Retrieved January 2013.
- ↑ Numbers from linear array notation | Steps Toward Infinity!
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) · diteral(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
RedBandServerYT's extensions: tridecalplex
Douglas Shamlin Jr.'s extensions: grand tridecalplex · quintuperdecal · triteral · dutetradecal · quadriteral · dupentadecal · duladecal · triladecal
MultiArrest91's extensions: tritredecal · tritetradecal · sextuperdecal · septuperdecal · octuperdecal · nonuperdecal · decuperdecal · centuperdecal · milluperdecal · quadritetradecal · quintitridecal
SeveralLegend9998's extensions: corboral
XRQ CORPORATION's extensions: grand tridecal · grand emperal · grand grand emperal · quintiteral · quadrilatri · quintilatri · sextilatri · septilatri · octilatri · nonilatri · decilatri
Exeen Management World's extensions: dutritet · dutetratet · dupentapent · duheptasept · duheptadecal · duoctadecal · duennadecal · grand dutritri · demperal · tritridecal · tritetradecal · quadritridecal
BAT4423's extensions: duhexadecal
ARsygo extensions: grand tetratri · grand tetradecal