Periodicity
When looking at the first row of entries in a Laver table, it can be seen that 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, ... (sequence A098820 in 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 (a large cardinal), it actually increases without bound. 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.
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