"Collatz 3n+1 conjecture is unprovable" paper; one last comment


A while back (Mon, May 15 2006 8:50 am), Craig Feinstein wrote in
sci.math:

if the Theorem 2 [in his paper] is wrong, then you should be able to
pick a number like 7 or 32 and rigorously prove that when applied to
the Collatz algorithm, the algorithm will halt at one *without talking
about what the algorithm does at each iteration*. If you can do this,
then and only then have you shown that Theorem 2 is wrong. If not, then
you are simply fooling yourself.

Searching through MathSciNet for a paper on Collatz and the 2-adics, I
found the following paper:

Stefan Andrei, Cristian Masalagiu, "About the Collatz conjecture."
Acta Informatica 35 (1998), no. 2, 167--179.

In the abstract, it states: "[The authors] show that the value of $k$
such that $f^{(k)}(n) =1$ can be found by an algorithm faster than the
one deriving from a direct application of the definition. They argue,
but don't give a definite proof, that their algorithm is three times
faster than the trivial one (the one obtained by applying the
definitions)."

So not only can it be proved that f^(k)(n) = 1 without using the
definition, there's actually a quicker way without the definition.

BTW, I can't find any references to whether the Collatz (3x+1) problem
is true for 2-adics, or what behavior occurs there, and MathSciNet
isn't that useful. Can anyone provide an article or a link? Thanks.

--- Christopher Heckman

.

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