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The tritet is equal to $$\lbrace 4,4,4 \rbrace = 4 \lbrace4\rbrace 4 = 4\uparrow\uparrow\uparrow\uparrow4$$ (4 hexated to 4) in BEAF.[1] The term was coined by Jonathan Bowers. This is the fourth Ackermann number.

Googology Wiki user Hyp cos calls this number a trientet, and it's equal to s(4,4,4), s(4,2,5), s(4,4,1,2), s(4,1,1,3), or s(4,1,1,1,2) in strong array notation. [2]

Computation Edit

Tritet can be computed in the following process:

• $$a_1 = 4$$
• $$a_2 = 4 \uparrow\uparrow 4 = 4 \uparrow\uparrow\uparrow 2$$ (this is tritet Jr.)
• $$a_3 = 4 \uparrow\uparrow a_2 = 4 \uparrow\uparrow\uparrow 3$$
• $$a_4 = 4 \uparrow\uparrow a_3 = 4 \uparrow\uparrow\uparrow 4$$
• etc.
• Tritet is equal to $$a_{a_{a_4}}$$.

Etymology Edit

The name of this number is based on tri- (three) and tetra- (four), meaning that there are 3 4's in the array.

ApproximationsEdit

Notation Approximation
Up-arrow notation $$4\uparrow\uparrow\uparrow\uparrow4$$ (exact)
BEAF $$\{4,4,4\}$$ (exact)
Hyper-E notation $$E(4)1\#1\#1\#4$$ (exact)
Chained arrow notation $$4\rightarrow4\rightarrow4$$ (exact)
Hyperfactorial array notation $$5!3$$
Fast-growing hierarchy $$f_5(3)$$
Hardy hierarchy $$H_{\omega^5}(3)$$
Slow-growing hierarchy $$g_{\varphi(3,0)}(4)$$

Sources Edit

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

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