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.