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# 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.

## Mathematics Subject Classification

05C38*no label found*

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## Comments

## Will gladly give up to Schneelocke

This entry was requested by Schneelocke in 2003. He hasn't logged on to PlanetMath since 2004, the same year he created this entry for Wikipedia. I've copied his entry over here. I will gladly give this one up to him if he comes back here.

Naturally, I'm skeptical of this topic, but the fact that it has an entry in the OEIS, and with an A number < 10^5 makes me a little less doubtful.