# Pentation

In mathematics, Pentation is the operation of repeated tetration, just as tetration is the operation of repeated exponentiation.[1]

Pentation can be written in Knuth's up-arrow notation as $a \uparrow \uparrow \uparrow b$ or $a \uparrow^{3}b$. In this notation, $a\uparrow b$ represents the exponentiation function $a^b$, which may be interpreted as the result of repeatedly applying the function $x\mapsto ax$, for $b$ repetitions, starting from the number 1. Analogously, $a\uparrow\uparrow b$, tetration, represents the value obtained by repeatedly applying the function $x\mapsto a\uparrow x$, for $b$ repetitions, starting from the number 1. And the pentation $a \uparrow \uparrow \uparrow b$ represents the value obtained by repeatedly applying the function $x\mapsto a\uparrow\uparrow x$, for $b$ repetitions, starting from the number 1.[2][3] Alternatively, in Conway chained arrow notation, $a\uparrow\uparrow\uparrow b = a\rightarrow b\rightarrow 3$.[4] 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.[citation needed]

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 $A(n,m)$ is defined by the Ackermann recurrence $A(m-1,A(m,n-1))$ with the initial conditions $A(1,n)=an$ and $A(m,1)=a$, then $a\uparrow\uparrow\uparrow b=A(4,b)$.[5]

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.[6]

## References

1. ^ Perstein, Millard H. (June 1962), "Algorithm 93: General Order Arithmetic", Communications of the ACM 5 (6): 344, doi:10.1145/367766.368160.
2. ^ 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.
3. ^ 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.
4. ^ Conway, John Horton; Guy, Richard (1996), The Book of Numbers, Springer, p. 61, ISBN 9780387979939.
5. ^ 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.
6. ^ Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic 12: 123–129, MR 0022537.