Properties of minimum of a loop

[roupam]

If a loop exists for positive numbers
then, if a is the minimum of all loops with n odds then,

a < n * 4^(n-1) for n > 1

I have a proof...
But I'd rather have some comments on the statement above...
Is it something interesting or worthwhile??

What I see is that, this formula can be used to check the existence of
a loop...

For example,

for an loop with n odd numbers
--> check whether all the odd numbers from 1 to n * 4^(n-1) contain a
loop of length n or not
--> if there isnt any, then no positive loop containing n odd numbers
exists

[Mensanator]

On Jun 12, 11:46�pm, roupam <roupam.gh...@gmail.com> wrote:
> If a loop exists for positive numbers
> then, if a is the minimum of all loops with n odds then,
>
> a < n * 4^(n-1) � � � � � � � for n > 1

Hey, you finally said something I don't have a quick
reply to!

>
> I have a proof...

I'll take that with a gain of salt.

> But I'd rather have some comments on the statement above...
> Is it something interesting or worthwhile??
>
> What I see is that, this formula can be used to check the existence of
> a loop...
>
> For example,
>
> for an loop with n odd numbers

You do realize that Lagarias proved a counterexample
must have at least 275000 elements? If he based that
on (3n+1)/2, then about 137500 of them would be odd.

> --> check whether all the odd numbers from 1 to n * 4^(n-1) contain a
> loop of length n or not

So, all I have to do is check 137500*4**137449 sequebces.
That number has 82789 digits. To get some idea of the
magnitude, see how long it would take simply to list
all of the numbers of 602 digits`(2000-bit):

http://mensanator.com/mensanator666/fun/2000_bit.htm

> --> if there isnt any,

I fear you'll find that approach a tad impractical.

> then no positive loop containing n odd numbers
> exists

And then you'll have to look for sequences with
137501 odds...

[roupam]

On Jun 13, 10:32 am, Mensanator <mensana...@aol.com> wrote:
> On Jun 12, 11:46 pm, roupam <roupam.gh...@gmail.com> wrote:
>
> > If a loop exists for positive numbers
> > then, if a is the minimum of all loops with n odds then,
>
> > a < n * 4^(n-1) for n > 1
>
> Hey, you finally said something I don't have a quick
> reply to!
>

Thanks... ;)

[jillbones]

On Jun 12, 10:32 pm, Mensanator <mensana...@aol.com> wrote:
> On Jun 12, 11:46 pm, roupam <roupam.gh...@gmail.com> wrote:
>
> > If a loop exists for positive numbers
> > then, if a is the minimum of all loops with n odds then,
>
> > a < n * 4^(n-1) for n > 1
>
> Hey, you finally said something I don't have a quick
> reply to!
>
>
>
> > I have a proof...
>
> I'll take that with a gain of salt.
>
> > But I'd rather have some comments on the statement above...
> > Is it something interesting or worthwhile??
>
> > What I see is that, this formula can be used to check the existence of
> > a loop...
>
> > For example,
>
> > for an loop with n odd numbers
>
> You do realize that Lagarias proved a counterexample
> must have at least 275000 elements?

Please explain how he was able to
establish such a proof?

>If he based that
> on (3n+1)/2, then about 137500 of them would be odd.

Actually, there must be more evens than
odds if a loop exists.

Bill J

[Mensanator]

On Jun 21, 1:42�pm, jillbones <b92...@yahoo.com> wrote:
> On Jun 12, 10:32�pm, Mensanator <mensana...@aol.com> wrote:
>
>
>
>
>
> > On Jun 12, 11:46 pm, roupam <roupam.gh...@gmail.com> wrote:
>
> > > If a loop exists for positive numbers
> > > then, if a is the minimum of all loops with n odds then,
>
> > > a < n * 4^(n-1) for n > 1
>
> > Hey, you finally said something I don't have a quick
> > reply to!
>
> > > I have a proof...
>
> > I'll take that with a gain of salt.
>
> > > But I'd rather have some comments on the statement above...
> > > Is it something interesting or worthwhile??
>
> > > What I see is that, this formula can be used to check the existence of
> > > a loop...
>
> > > For example,
>
> > > for an loop with n odd numbers
>
> > You do realize that Lagarias proved a counterexample
> > must have at least 275000 elements?
>
> Please explain how he was able to
> establish such a proof?

_I_ can't.

I'm refering to the Mathworld artilce
<quote>
Lagarias (1985) showed that there are no nontrivial
cycles with length <275000.
</quote>

I don't have that paper. That's why I don't know if
"cycles" refers to using the (3n+1)/2 option.

>
> >If he based that
> > on (3n+1)/2, then about 137500 of them would be odd.
>
> Actually, there must be more evens than
> odds if a loop exists.

Sure, but how much more? Statistically, it's
about two evens per odd. But if you're using the
(3n+1)/2 option, one of the factors of two is already
accounted for, so the number of odds would be either
1/2 or 1/3 of 275000.

Of course, reality doesn't always match statistics,
especially when dealing with resonators, but that's
the way to bet since you're more likely to encounter
random bit patterns that resonators.