Necessary condition for a loop

[roupam]

A loop a(1), a(2), ... , a(n)
must satisfy the following...
if a sequence doesnt satisfy the following then it is not a loop...

c * t < (3s + n) p

where
c = (3^n-1 + 3^n-2 * 2^S(1) + ... + 3 * 2^S(n-2) + 2^S(n-1))
t = Sum( 2^k(i) ) for i = 1 to n
s = Sum( a(i) ) i = 1 to n
p = 2^S(n) - 3^n

k(i) is the highest power of 2 that divides 3a(i)+1


Roupam

[roupam]

and S(i) = k(1) + k(2) + ... + k(i)

[Mensanator]

On Jun 7, 8:00�pm, roupam <roupam.gh...@gmail.com> wrote:
> A loop a(1), a(2), ... , a(n)
> must satisfy the following...
> if a sequence doesnt satisfy the following then it is not a loop...
>
> c * t < (3s + n) p

The real answer is that to be a loop, it suffices that

c == 0 (mod p)

BTW, you *DO* know that in the rationals, *EVERY*
sequence is a loop cycle, don't you? It's when the
rational cycle point reduces to an interger that the
same seuence is also a loop in Collatz.

[roupam]

> > A loop a(1), a(2), ... , a(n)
> > must satisfy the following...
> > if a sequence doesnt satisfy the following then it is not a loop...
>
> > c * t < (3s + n) p
>
> The real answer is that to be a loop, it suffices that
>
> c == 0 (mod p)

Thats a condition where i started from...
But that condition doesnot limit itself to what kind of numbers are
present in the loop... (i mean to say whether they are integers or
not)

> BTW, you *DO* know that in the rationals, *EVERY*
> sequence is a loop cycle, don't you? It's when the
> rational cycle point reduces to an interger that the
> same seuence is also a loop in Collatz.

Yes

[Mensanator]

On Jun 8, 2:40 am, roupam <roupam.gh...@gmail.com> wrote:
> > > A loop a(1), a(2), ... , a(n)
> > > must satisfy the following...
> > > if a sequence doesnt satisfy the following then it is not a loop...
>
> > > c * t < (3s + n) p
>
> > The real answer is that to be a loop, it suffices that
>
> > c == 0 (mod p)
>
> Thats a condition where i started from...
> But that condition doesnot limit itself to what kind of numbers are
> present in the loop... (i mean to say whether they are integers or
> not)

True, it doesn't limit itself, it works for rationals that
aren't integers just as well.

BUT, if you start with an interger (that's given, isn't it?),
then since, via the Collatz rules, the successor of an odd
integer is an even integer and the successor of an even integer
is an integer (even or odd), your concern is moot. You do NOT
need to know whether the values are integers or not, it follows
from the Collatz rules that they will ALWAYS be integers.

Also, you depend on knowing what the actual values are
(a(1), a(2)...). This is completely unnessessary. First of all,
if you have the values, you already know whether it's a loop, the
inequality is unneccessary.

More importantly, a loop is solely determined by k(1), k(2)...
You can determine loop cycles without ever knowing what the
values in the loop are. It is the STRUCTURE of the Collatz
sequence that determines loops, not the values (keep in mind
that every possible structure is found infinitely many times
on the Collatz graph, but there can only be one that's a loop
cycle). What would you prefer, checking an infinite number of
cases for an infinite number of structures, or just one case
for each structure (well, it's infinite either way, so no brute
force, but the second is still preferable).

Once you've found a loop, you can then determine the values,
but the values are not necessary for finding the loop.

Value-centric thinking will always lead you astray. It is
structure-centric thinking that you want to embrace. Think
Trivial vs non-Trivial loop cycles. The Trivial positve loop
cycle is 4C, 2C, C for 3n+C, 4, 2, 1 is just the value-centric
case that applies ONLY when C=1. Do your proofs using algebra,
with variables. When you get that to work, plug in some numbers
to make make sure it all hangs together, but that should be
a check on the proof, not the proof itself.