Re: seeking simple integer number 'games'

The Qurqirish Dragon wrote:
> Well, starting at -17:
> -17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182,
> -91, -272, -136, -68, -34, -17
>
> So this is a cycle of length 18.
>
> However, If the original conjecture pertained to positive integers
> only, then finding a negative cycle has no bearing on the validity of
> the conjecture.

Actually, it does. Because cycles are determined solely by
the pattern of [3x+1],[x/2] operations. From every sequence of
such operations, one can derive a Hailstone Function

X*a - Z*C
g = -----------
Y

where 'a' and 'g' are the ending and starting points of the
sequence and must be integers. X, Y, Z are constants based
on the actual pattern of [3x+1],[x/2] operations. C is used
to generalize the system to 3x+C.

The Hailstone Function is the equation of a straight line
whose slope is X/Y. Because this slope can never be 1
(X being a power of 2 and Y a power of 3), it cannot be
parallel to the identity line g = a and so must intercept
it.

It intercepts it at

Z * C
CP = -------
X - Y

Note that both 'g' and 'a' have dropped out. The cycle point
is determined solely by the sequence pattern and is independent
of whether 'a' and 'g' are in the positive or negative domain.

> If you can prove that the conjecture is true on a
> superset of the positive integers, then you have proven it, but showing
> it doesn't work on a superset does NOT make it false.

That's my 'spin' on the problem. I'm saying that the integers
are not a superset but that the positive numbers are a subset
and that to understand why it is true for the subset, one must
understand the full set. And it just happens to be false for
the full set.

Sure, it may be true for the subset. But since cycles are
independent of the domain, how would you prove there are
no non-trivial cycles in the positive domain (which is made
particularly hard by the fact you cannot prove there are no
non-trivial cycles).

>
> I just checked :
> http://en.wikipedia.org/wiki/Collatz_conjecture
>
> For a statement of the conjecture, and it appears to pertain only to
> positive integers. And the claim is that the sequence will always,
> eventually, hit 1.

And you can create a whole family of systems by generalizing
3x+1 to 3x+C. Then you must restate the conjecture as "will
always, eventually, hit C". People have actually gotten
confused about that, thinking that 3x+3 or 3x+5 must also
eventually hit 1 which is not the case.

And it is a special case for 3x+1 that sequences cannot cross
domains. If you start in the negative domain you stay in the
negative domain. But that's not true in the general case.

For example, in 3x+5 you get crossover:

-8 -4 -2 -1 +2 +1 +8 +4 +2 +1

>
> I have not searched for an "official" source, so take my link with the
> required care that any wikipedia entry should have.

It would be so much nicer if the conjecture was generally true.
It can still be specifically true even though generally false,
but it will certainly be a lot harder to prove.

.

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