Recurring digits in tetration and the Ackermann function

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Daniel Geisler

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Mar 24, 2007, 5:30:04 PM3/24/07
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Greetings,
I am finishing writing a paper on recurring digits in tetration and
the Ackermann function inspired
by a problem in a number theory class I'm taking, but I am at a lose
for references.

Consider the tetrates of 3 - 3, 27, 7625597484987, ...; beginning with
the third tetrate of 3, all
tetrates of 3 end in the digits 87. This is a consequence of the Euler
Phi function or the totient
for powers. Below 3^(10^2) = ...621272702107522001 is shorthand for
the fact that
3^(10^2) is congruent to 621272702107522001 (mod 10^18). The process
of iterated exponentiation
acts to pump the entropy out of the least significant digits until
each successive exponentiation
"freezes" the next unfrozen least significant digit. I would expect
that there should be a reference
to the following phenomena somewhere:

3^(10^0) = 3
3^(10^1) = 59049
3^(10^2) = ...621272702107522001
3^(10^3) = ...102768902855220001
3^(10^4) = ...498105206552200001
3^(10^5) = ...250669865522000001
3^(10^6) = ...468478655220000001
3^(10^7) = ...862786552200000001

I doubt the following has any references, but just in case,
for n >= 10 and using ^^ to denote tetration:

2^^n = ...2948736
3^^n = ...4195387
4^^n = ...1728896
5^^n = ...8203125
6^^n = ...7238656
7^^n = ...5172343
8^^n = ...5225856
9^^n = ...2745289
11^^n = ...2666611
12^^n = ...4012416
13^^n = ...5045053
14^^n = ...7502336
15^^n = ...0859375
16^^n = ...0415616
17^^n = ...0085777
18^^n = ...4315776
19^^n = ...9963179

Just as each tetrate is also power, all numbers generated by the
higher arithmetic operators beyond
tetration are also tetrates. Therefore the phenomena of the freezing
of least significant digits
occurs in all arithmetic operators beyond exponentiation.
Thanks,
Daniel

Oscar Lanzi III

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Mar 25, 2007, 12:30:03 PM3/25/07
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Another way to put this is to say that the tetration function converges
p-adically.

You might want to investigate what happens in base 10 with the series 6,
6^5, 6^5^5, etc.; and with 5, 5^2, 5^2^2, etc.

--OL

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