In mathematics, Laver tables (named after mathematician Richard Laver, who discovered them towards the end of the 1980s in connection with his works on set theory) are tables of numbers that have certain properties.
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For a given a natural number n, one can define the nth Laver table (with 2^{n} rows and columns) by setting
where p denotes the row and q denotes the column of the entry. We define
and then continue to calculate the remaining entries of each row from the mth to the first using the equation
The resulting table is then called the nth Laver table; for example, for n = 2, we have:
1  2  3  4  
1  2  4  2  4 
2  3  4  3  4 
3  4  4  4  4 
4  1  2  3  4 
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.
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, ... The sequence is increasing, and it was proved in 1995 by Richard Laver that under the assumption that there exists a rankintorank, 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.
