Tritri

The tritri is equal to \(\{ 3,3,3 \} = 3 \{3\} 3\) (3 pentated to 3) \(= 3 \{2\} 7625597484987\) in BEAF. It is the third Ackermann Number. It is equal to the power tower with 7625597484987 3's high. In contrast, a power tower of only four threes is between googol and googolplex.

Tritri is too large to represent anything in known world. Even the largest number ever appeared in physics, an estimate of the time when the Universe returns to the same state, well less even than \(3 \uparrow\uparrow 7\).

Tritri decomposes to \(3 \uparrow\uparrow 3 \uparrow\uparrow 3\).

Jonathan Bowers, who coined the name, has created many other googologisms based on the number 3 (such as ultatri and triakulus). This is because 3 is the smallest positive integer that does not create degenerate arrays like 1 and 2, since \(\{ 2,2,n \} = 4\) for all \(n > 0\).

The last 10 digits of tritri are ...2464195387.

Milton Green proved that tritri is less than.

Computation
Tritri can be computed in the following process:


 * \(a_1 = 3\)
 * \(a_2 = 3^3 = 27\)
 * \(a_3 = 3^{3^3} = 7625597484987\)
 * \(a_4 = 3^{3^{3^3}}\) (a number with trillions of digits)
 * etc.
 * Tritri is equal to \(a_{7625597484987}\).

Etymology
The name of this number is based on the word "tri-" (greek: three).