# Where in the fast growing hierarchy is the next level beyond Latri

big numberselementary-number-theoryreference-request

I found the huge number Latri here

If I understand it right, it can be written as

$$< 3 , 3 , 3 >$$

$$< 2 >$$

with Bowers 2-dimensional arrays.

Is this correct ?

I wonder where the number

$$< 3 , 3 , 3 >$$

$$< 3 >$$

lies in the fast growing hierarchy. It should be VASTLY larger then Latri, but how much larger ?
It must be far below $$f_{\omega^{\omega^2}}(3)$$ which is approximately the magnitude of

$$< 3 , 3 , 3 >$$

$$< 3 , 3 , 3 >$$

$$< 3 , 3 , 3 >$$

Does someone know a good approximation in the fast growing hierarchy ?

Latri is $$\{3,3,3\;(1)\;2\}$$ which, as you correctly stated, represents a two-row array with $$\{3,3,3\}$$ in the first row and a sole $$\{2\}$$ in the second row.

The second number you've mentioned can be written as $$\{3,3,3\;(1)\;3\}$$.

In general, $$\{a,b,c\;(1)\;x\}$$ is in the same ballpark as $$f_{\omega^\omega \cdot (x-1)+(c-1)}(b)$$ (the value of $$a$$ doesn't change the final value by much).

So:

$$\{3,3,3\;(1)\;2\} \approx f_{\omega^\omega+2}(3)$$ (that's the Latri)

$$\{3,3,3\;(1)\;3\} \approx f_{\omega^\omega \cdot 2+2}(3)$$