# 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 $i\star 1=(imod{2}^{n})+1$ 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

$$\left[\begin{array}{cccc}\hfill 2\hfill & \hfill 4\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 4\hfill & \hfill 4\hfill & \hfill 4\hfill & \hfill 4\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$$ |

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 |
---|---|

Canonical name | LaverTable |

Date of creation | 2013-03-22 16:26:13 |

Last modified on | 2013-03-22 16:26:13 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 05C38 |