Collatz-Intro - The loop-problem
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The
loop-problem is the question, whether there exists any transformation with
finite many exponents A,B,C where the output-number a' equals the input-number
a: That means
for a transformation a'=
T(a;A,B,C,...H) that a'
equals a, where a is an odd number>1. The latter
restriction is made, since there is the "trivial" loop 1 =
T(1;2,2,2,2...2) for any number N of
exponents "2" (and other loops,
if negative "a" are allowed: -1 = T(-1;1,1,1,1...1) for any number N of exponents "1" the loop is -1 -> -1 -> ... -5 = T(-5;1,2,1,2,1,2...1,2) for any number N of pairs of exponents
"1,2" the loop is -5 -> -7 -> -5 -> ... -17= T(-17;1,1,1,2,1,1,4,..) the loop is -17 -> -25 -> -37 -> -55 -> -41 -> -61 -> -91
-> -17... ) In the following
I assume only positive numbers, and I
may omit the clause "... except the trivial one..." for
readability. |
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It is very
simple to prove, that there cannot be another loop of length 1 except the
trivial. This is shown here to introduce the reader to my concept and the main
formula, how to disprove the possibility of a loop in general without much formalism
around. Also it is
possible to exclude certain loops up to lengthes around 100..200 steps (which
means about 200...300 steps of the original definition of the collatz-transformation)
by an elementary proof using obvious bounds for the exponents. Useful
results are already known in the collatz-research. I mean for instance such
results from using approximation-tools, or things like: since we know, all
numbers below 2^40 are known not to belong to a loop, a loop - if it exists
must have the minimum length of 250000 steps or the like. The interested
reader may visit the [Rosendaal]-site for up-to-date results. Since I were
unable to contribute to the mentioned research, but more, because my interest
is more an analytical approach, I never entered this path of search. Even
more, it seemed to be a promising
attack, to use a sophisticated system of modular classes, which had already
helped me in understanding some interesting rules of transforms. This means
to eventually disprove the possibility of such a looping with arguments of
modular arithmetics. This was backed from the canonical formula for a loop,
since a loop is disproven for such cases, where nominator and denominator
cannot not be divided without remainder. But unfortunately this problem could
not be solved yet despite some good progress in the beginning, and was therefore
put away awaiting a later consideration. Since the
general loop (with arbitrary exponents) could not successfully be attacked
this way so far, I decided, first to study a simple form of a loop, such that
it is constructed by several ascending steps, and then descends
with one single additional step. I called this type of loop
"primitive loop" and got very far with a disproof. Recently I
found an article, that just this "primitive-loop" was successfully
attacked by Ray Steiner already in 1978, who used the name "1-cycle"
for that construct, and who was able to disprove the possibility of such a
loop with results from the theory of rational approximation with transcendent
numbers. But since that result requires some deep insight in this theory, I
continued my try to find a more elementary proof. There exist
(numerical) disproofs for "general loops" up to the length of
250000, as I stated above, disproofs for "primitive-loops" of
arbitrary lengthes, and even for loops consisting of concatenated
primitive-transformations, called "m-cycle" up to the
number of 68 concatenates (which also includes, that each involved partial
primitive transformation is of arbitrary length) |
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It is also
worth to be noted, that all those approaches to the disproof of a loop must
somehow be aware of deep connections between the involved parameters 3/2/1 of
the collatz-transformation, since the analoguous stated 5x+1-problem actually
has loops, and even short ones, and also primitive loops. Keeping the
equivalent formulae of that 5x+1-problem in mind sometimes helped to prevent
to try too general formulae and to return to the specifics of the
3/2/1-relation. Especially in
the question of the "primitive loop" (1-cycle) very tight
connections to some unsolved general questions related to the ratio of
logarithms of 2 and 3 are obvious, so it is in any case worth to watchout the
proceedings in that area of research. |
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last update: 25.7.2005€ |
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