Tritri

Tritri is equal to \(3 \uparrow\uparrow\uparrow\uparrow 2 = 3 \uparrow\uparrow\uparrow 3\) (3 pentated to 3) \(= 3 \uparrow\uparrow\) in up-arrow notation. In BEAF it can be written as {3,3,3}, {3,3(1)2} or {3,2/2}. It is the third Ackermann Number. 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.

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

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 over 3.6 trillion digits)
 * etc.
 * Tritri is equal to \(a_{a_3} = a_{7625597484987}\).

Size
The number is equal to a power tower of 3's, 7625597484987 levels high. In contrast, a power tower of only four threes is between googol and googolplex, and \(3 \uparrow\uparrow 7\) is already larger than the Poincare recurrence time, often touted as the largest number to appear in physics. It is much smaller than Graham's number, on the other hand.

Milton Green proved that tritri is less than.

On the group blog LessWrong, the following question was proposed as a moral thought experiment: "Would you prefer that one person be horribly tortured for fifty years without hope or rest, or that 3↑↑↑3 people get dust specks in their eyes?"

Etymology
The name of this number is based on the word "tri-" (Greek for three), meaning a size-3 linear array of 3's.