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The first three values of the expression x[5]2. The value of 3[5]2 is about 7.626 × 1012; values for higher x, such as 4[5]2, which is about 2.361 × 108.072 × 10153 are much too large to appear on the graph.

In mathematics, pentation (or hyper-5) is the next hyperoperation after tetration and before hexation. It is defined as iterated (repeated) tetration, just as tetration is iterated exponentiation.[1] It is a binary operation defined with two numbers a and b, where a is tetrated to itself b-1 times. For instance, using hyperoperation notation for pentation and tetration, means tetrating 2 to itself 2 times, or . This can then be reduced to


The word "pentation" was coined by Reuben Goodstein in 1947 from the roots penta- (five) and iteration. It is part of his general naming scheme for hyperoperations.[2]


There is little consensus on the notation for pentation; as such, there are many different ways to write the operation. However, some are more used than others, and some have clear advantages or disadvantages compared to others.

  • Pentation can be written as a hyperoperation as . In this format, 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.[3][4] This will be the notation used in the rest of the article.
  • In Knuth's up-arrow notation, is represented as or . In this notation, represents the exponentiation function and represents tetration. The operation can be easily adapted for hexation by adding another arrow.
  • In Conway chained arrow notation, .[5]
  • Another proposed notation is , though this is not extensible to higher hyperoperations.[6]


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 .[7]

As tetration, its base operation, has not been extended to non-integer heights, pentation is currently only defined for integer values of a and b where a > 0 and b ≥ −2, and a few other integer values which may be uniquely defined. As with all hyperoperations of order 3 (exponentiation) and higher, pentation has the following trivial cases (identities) which holds for all values of a and b within its domain:

Additionally, we can also define:

Other than the trivial cases shown above, pentation generates extremely large numbers very quickly such that there are only a few non-trivial cases that produce numbers that can be written in conventional notation, as illustrated below:

  • (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note )
  • (a number with over 10153 digits)
  • (a number with more than 10102184 digits)

See also


  1. ^ Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic", Communications of the ACM, 5 (6): 344, doi:10.1145/367766.368160, S2CID 581764.
  2. ^ Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic, 12 (4): 123–129, doi:10.2307/2266486, JSTOR 2266486, MR 0022537, S2CID 1318943.
  3. ^ Knuth, D. E. (1976), "Mathematics and computer science: Coping with finiteness", Science, 194 (4271): 1235–1242, Bibcode:1976Sci...194.1235K, doi:10.1126/science.194.4271.1235, PMID 17797067, S2CID 1690489.
  4. ^ 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 0549780.
  5. ^ Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
  6. ^ " - Tetration". Retrieved 2022-09-12.
  7. ^ 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.