# Laver table

A Laver table $L_{n}$ for a given integer $n>0$ has $2^{n}$ rows $i$ and columns $j$ with each entry being determined thus: $L_{n}(i,j)=i\star j$, with $i\star 1=(i\mod 2^{n})+1$ for the first column. Subsequent rows are calculated with $i\star(j\star k):=(i\star j)\star(i\star k)$.

For example, $L_{2}$ is

 $\begin{bmatrix}2&4&2&4\\ 3&4&3&4\\ 4&4&4&4\\ 1&2&3&4\\ \end{bmatrix}$

There is no known closed formula to calculate the entries of a Laver table directly, and it is in fact suspected that such a formula does not exist.

The entries repeat with a certain periodicity $m$. This periodicity is always a power of 2; the first few periodicities are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16, … (see A098820 in Sloane’s OEIS). The sequence is increasing, and it was proved in 1995 by Richard Laver that under the assumption that there exists a rank-into-rank, it actually tends towards infinity. Nevertheless, it grows extremely slowly; Randall Dougherty showed that the first $n$ for which the table entries’ period can possibly be 32 is $A(9,A(8,A(8,255)))$, where $A$ denotes the Ackermann function.

## References

• 1 P. Dehornoy, ”Das Unendliche als Quelle der Erkenntnis”, Spektrum der Wissenschaft Spezial 1/2001: 86 - 90
• 2 R. Laver, ”On the Algebra of Elementary Embeddings of a Rank into Itself”, Advances in Mathematics 110 (1995): 334

This entry based entirely on a Wikipedia entry from a PlanetMath member.

Title Laver table LaverTable 2013-03-22 16:26:13 2013-03-22 16:26:13 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 05C38