Editing Laver table (section)

== Are the first-row periods unbounded? ==

Looking at just the first row in the ''n''-th Laver table, for ''n'' = 0, 1, 2,&nbsp;..., the entries in each first row are seen to be periodic with a period that's always a power of two, as mentioned in Property&nbsp;2 above. The first few periods are 1, 1, 2, 4, 4, 8, 8, 8, 8, 16, 16,&nbsp;... {{OEIS|A098820}}. This sequence is nondecreasing, and in 1995 Richard Laver [[mathematical proof|proved]], ''under the assumption that there exists a [[rank-into-rank]] (a [[large cardinal]] property)'', that it actually increases without bound. (It is not known whether this is also provable in [[ZFC]] without the additional large-cardinal axiom.)<ref>{{citation
 | last = Laver | first = Richard
 | doi = 10.1006/aima.1995.1014 | doi-access=free
 | issue = 2
 | journal = [[Advances in Mathematics]]
 | mr = 1317621
 | pages = 334–346
 | title = On the algebra of elementary embeddings of a rank into itself
 | volume = 110
 | year = 1995
 | hdl = 10338.dmlcz/127328
 | hdl-access = free
 }}.</ref> In any case, it grows extremely slowly; Randall Dougherty showed that 32 cannot appear in this sequence (if it ever does) until ''n'' > A(9,&nbsp;A(8,&nbsp;A(8,&nbsp;254))), where A denotes the [[Ackermann function|Ackermann–Péter function]].<ref>{{citation
 | last = Dougherty | first = Randall |author-link=Randall Dougherty 
 | arxiv = math.LO/9205202
 | doi = 10.1016/0168-0072(93)90012-3
 | issue = 3
 | journal = Annals of Pure and Applied Logic
 | mr = 1263319
 | pages = 211–241
 | title = Critical points in an algebra of elementary embeddings
 | volume = 65
 | year = 1993
 | s2cid = 13242324 }}.</ref>