Laver table

A Laver table Ln for a given integer n>0 has 2n rows i and columns j with each entry being determined thus: Ln(i,j)=ij, with i1=(imod2n)+1 for the first column. Subsequent rows are calculated with i(jk):=(ij)(ik).

For example, L2 is


There is no known closed formula to calculate the entries of a Laver table directly, and it is in fact suspected that such a formulaMathworldPlanetmathPlanetmath 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 assumptionPlanetmathPlanetmath that there exists a rank-into-rank, it actually tends towards infinityMathworldPlanetmath. 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 functionMathworldPlanetmath.


  • 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