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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.

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

Mar 25, 2007, 12:30:03 PM3/25/07

to

Another way to put this is to say that the tetration function converges

p-adically.

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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