## Tuesday, August 14, 2018

### HP Prime and TI-84 Plus CE: Tetration, Iterated Exponentiation

HP Prime and TI-84 Plus CE: Tetration, Iterated Exponentiation

Introduction

Tetration is iterated exponentiation.  A common notation of tetration is the use of two upward arrows, known as Knuth’s up-arrow notation.  In general:

x y = x ^ x ^ x ^ … ^ x   (y times)

Take the x to its own power y times.

For example:

2 2 = 2 ^ 2 ^ 2 = 2 ^ 4 = 16

3 2 = 3 ^ 3 ^ 3 = 3 ^ 27 = 7625597484987

4 2 = 4 ^ 4 ^ 4 = 4 ^ 256 =
13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
≈ 1.34078078079299 * 10^154

First, thank goodness that the HP Prime can handle really long integers in CAS mode.  Second, you can quickly see how fast the results grow in tetration calculations.

In order to allow for a larger set of calculations, the programs are provided, where we break down the mantissa and exponents of each result.

HP Prime Program TETRATION

EXPORT TETRATION(X,Y)
BEGIN
// 2018-08-14 EWS
LOCAL I,M,E,S;
// X^^Y, Y is an integer
M:=MANT(X);
E:=XPON(X);
FOR I FROM 1 TO Y DO
S:=M*ALOG(E)*LOG(X);
M:=ALOG(FP(S));
E:=IP(S);
END;
RETURN {M,E};
END;

TI-84 Plus CE Program TETRATION

"EWS 2018-08-14"
Disp "TETRATION X^^Y","Y: INTEGER"
Prompt X,Y
10^(fPart(X))→M
iPart(log(X))→E
For(I,1,Y)
M*10^(E)*log(X)→S
10^(fPart(S))→M
iPart(S)→E
End
Disp M,"*10^",E

Source:

“Knuth’s Up-Arrow Notation” Wikipedia.  Last edited August 9, 2018.  Retrieved August 14, 2018.  https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

Eddie

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