Tripent visualized using iterated power towers.

Tripent with arrows

"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:

  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(A(A(.........A(A(A(A(1)))).........)))))), where there are A(A(A(A(A(......A(A(A(1)))......))))) A's, where there are A(A(A(A(......A( A(A(1)))......)))) A's, where there are A(A(A(A(A(1))))) A's.

Etymology Edit

The name of this number is based on tri- (three) and penta- (five), meaning that there are 3 5's in the array.


Notation Approximation
Up-arrow notation \(5\uparrow\uparrow\uparrow\uparrow\uparrow5\) (exact)
BEAF \(\{5,5,5\}\) (exact)
Hyper-E notation \(E(5)1\#1\#1\#1\#5\) (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 Edit

  1. Bowers, JonathanInfinity Scrapers. Retrieved January 2013.
  2. Numbers from linear array notation | Steps Toward Infinity!

See also Edit

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