Tripent visualized using iterated power towers.

"Fashion" view of tripent using iterated power towers with arrows.

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:

  1. \(A(1)\) = 5
  2. \(A(2)\) = 5^5^5^5^5 (A(1) 5's)
  3. \(A(3)\) = 5^5^5^...5^5^5, with A(2) 5's.
  4. \(A(4)\) = 5^5^5^5^5^......5^5^5^5, with A(3) 5's
  5. Continue, let \(A(n)\) be 5^5^5^5^......^5^5^5 with \(A(n-1)\) 5's
  6. 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)
BEAF \(\{5,5,5\}\) (exact)
Hyper-E notation \(\textrm E[5]1\#1\#1\#1\#5\) (exact)
Chained arrow notation \(5\rightarrow5\rightarrow5\) (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

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

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