Pentation can be written in hyperoperation with , or Knuth's up-arrow notation as or . In this notation, represents the exponentiation function , which may be interpreted as the result of repeatedly applying the function , for repetitions, starting from the number 1. Analogously, , tetration, represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. And the pentation represents the value obtained by repeatedly applying the function , for repetitions, starting from the number 1. Alternatively, in Conway chained arrow notation, . Until a consensus for the definition of tetration to real heights is reached, pentation can't even be defined to an integer exponent higher than 1 when its base is not a whole number.
The values of the pentation function may also be obtained from the values in the fourth row of the table of values of a variant of the Ackermann function: if is defined by the Ackermann recurrence with the initial conditions and , then .
- Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic", Communications of the ACM 5 (6): 344, doi:10.1145/367766.368160.
- Knuth, D. E. (1976), "Mathematics and computer science: Coping with finiteness", Science 194 (4271): 1235–1242, doi:10.1126/science.194.4271.1235, PMID 17797067.
- Blakley, G. R.; Borosh, I. (1979), "Knuth's iterated powers", Advances in Mathematics 34 (2): 109–136, doi:10.1016/0001-8708(79)90052-5, MR 549780.
- Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
- Nambiar, K. K. (1995), "Ackermann functions and transfinite ordinals", Applied Mathematics Letters 8 (6): 51–53, doi:10.1016/0893-9659(95)00084-4, MR 1368037.
- Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic 12: 123–129, MR 0022537.