Richard Laver proved that there is a unique binary operation $\*$ on $\{1,\ldots,2^n\}$ which satisfies $$a\*1 \equiv a+1 \mod 2^n$$
$$a\* (b\* c) = (a\* b) \* (a \* c).$$
This is the $n$th Laver table $(A\_n,\*)$.


There is an algorithm for computing $a \* b$ in $A\_n$, but in general (and especially for small values of $a$), this requires one to compute much of the rest of $A\_n$.
What is the largest value for $n$ for which someone can, in a modest amount of time, compute an arbitrary entry in $A\_n$? I am able to compute entries in $A\_{27}$.


I should note that the map which sends $a$ to $a\ \mathrm{mod}\ 2^m$ defines a homomorphism from $A\_n$ to $A\_m$ for $m < n$ and hence the problem becomes strictly harder for larger $n$.


Edit: I have actually been able to compute $A\_{28}$, not just $A\_{27}$.



What is the largest Laver table which has been computed?
I've been in contact with Patrick Dehornoy and Ales Drapal and both thought that $A\_{28}$ is likely the current record for a Laver table computation.


