Hyperoperation

Hyperoperation

In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations) that starts with the unary operation of successor, then continues with the binary operations of addition, multiplication and exponentiation, after which the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration, pentation) and can be written using n-2 arrows in Knuth's up-arrow notation (if the latter is properly extended to negative arrow-indices for the first three hyperoperations). Each hyperoperation may be understood recursively in terms of the previous one by:

with b occurrences of a on the right hand side of the equation

It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:

This recursion rule is common to many variants of hyperoperations (see below).