Pentation

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

History

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]

Notation

Pentation can be written as a hyperoperation as ${\displaystyle a[5]b}$, or using Knuth's up-arrow notation as ${\displaystyle a\uparrow \uparrow \uparrow b}$ or ${\displaystyle a\uparrow ^{3}b}$. In this notation, ${\displaystyle a\uparrow b}$ represents the exponentiation function ${\displaystyle a^{b}}$ or ${\displaystyle a[3]b}$, which may be interpreted as the result of repeatedly applying the function ${\displaystyle x\mapsto a[2]x}$, for ${\displaystyle b}$ repetitions, starting from the number 1. Analogously, ${\displaystyle a[4]b}$, tetration, represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a[3]x}$, for ${\displaystyle b}$ repetitions, starting from the number 1. And the pentation ${\displaystyle a[5]b}$ represents the value obtained by repeatedly applying the function ${\displaystyle x\mapsto a[4]x}$, for ${\displaystyle b}$ repetitions, starting from the number 1.[3][4] Alternatively, in Conway chained arrow notation, ${\displaystyle a[5]b=a\rightarrow b\rightarrow 3}$.[5] Another proposed notation is ${\displaystyle {_{b}a}}$, though this is not extensible to higher hyperoperations.[6]

Examples

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 ${\displaystyle A(n,m)}$ is defined by the Ackermann recurrence ${\displaystyle A(m-1,A(m,n-1))}$ with the initial conditions ${\displaystyle A(1,n)=an}$ and ${\displaystyle A(m,1)=a}$, then ${\displaystyle a[5]b=A(4,b)}$.[7]

As its base operation (tetration) has not been extended to non-integer heights, pentation ${\displaystyle a[5]b}$ is currently only defined for integer values of a and b where a > 0 and b ≥ −1, and a few other integer values which may be uniquely defined. Like all other 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:

• ${\displaystyle 1[5]b=1}$
• ${\displaystyle a[5]1=a}$

• ${\displaystyle a[5]0=1}$
• ${\displaystyle a[5](-1)=0}$

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:

• ${\displaystyle 2[5]2=2[4]2=2^{2}=4}$
• ${\displaystyle 2[5]3=2[4](2[4]2)=2[4]4=2^{2^{2^{2}}}=2^{2^{4}}=2^{16}=65,536}$
• ${\displaystyle 2[5]4=2[4](2[4](2[4]2))=2[4](2[4]4)=2[4]65536=2^{2^{2^{\cdot ^{\cdot ^{\cdot ^{2}}}}}}{\mbox{ (a power tower of height 65,536) }}\approx \exp _{10}^{65,533}(4.29508)}$ (shown here in iterated exponential notation as it is far too large to be written in conventional notation. Note ${\displaystyle \exp _{10}(n)=10^{n}}$)
• ${\displaystyle 3[5]2=3[4]3=3^{3^{3}}=3^{27}=7,625,597,484,987}$
• ${\displaystyle 3[5]3=3[4](3[4]3)=3[4]7,625,597,484,987=3^{3^{3^{\cdot ^{\cdot ^{\cdot ^{3}}}}}}{\mbox{ (a power tower of height 7,625,597,484,987) }}\approx \exp _{10}^{7,625,597,484,986}(1.09902)}$
• ${\displaystyle 4[5]2=4[4]4=4^{4^{4^{4}}}=4^{4^{256}}\approx \exp _{10}^{3}(2.19)}$ (a number with over 10153 digits)
• ${\displaystyle 5[5]2=5[4]5=5^{5^{5^{5^{5}}}}=5^{5^{5^{3125}}}\approx \exp _{10}^{4}(3.33928)}$ (a number with more than 10102184 digits)

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. ^ Goodstein, R. L. (1947), "Transfinite ordinals in recursive number theory", The Journal of Symbolic Logic, 12: 123–129, MR 0022537.
3. ^ 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.
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. ^ http://www.tetration.org/Tetration/index.html
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.