Pentation

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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]

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

↑ 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: 1235–1242, doi:10.1126/science.194.4271.1235 .
↑ 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 .

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