[Coco] 5 Simple Math Problems No One Can Solve (Kip Koon)

[Hugo Dufort]

Hi Arthur and Dave,
Any proof needs to be formal, though some proofs involve some brute 
force. For example, if someone were to show that all existing cases can 
be simplified into a subset of 500 000 root examples, then solving these 
500 000 root examples would constitute a proof. It may look weird but 
I've seen some proofs in Set Theory that go like this.

However this would not work for the Collatz conjecture.

I won't give all the details of my "incomplete" proof, but I'll just say 
it involves working around the junction points (where 2 branches 
converge) and deriving a new set of functions. These functions are 
easier to work with, because they have a standardized form -- they're 
Collatz-like, and proving that they always converge proves that Collatz 
always converge. From this point, you dive into some fractal-like 
weirdness involving resonant functions.

------------------------------
> Message: 3
> Date: Wed, 19 Oct 2016 18:53:00 -0400
> From: Arthur Flexser <flexser at fiu.edu>
> To: CoCoList for Color Computer Enthusiasts <coco at maltedmedia.com>
> Subject: Re: [Coco] 5 Simple Math Problems No One Can Solve (Kip Koon)
> Message-ID:
> 	<CA+LuDcfA=dXLGcaX_QO4rp4bcGmcVeKppFPt4Yf+757gOxGMEg at mail.gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> Wikipedia makes this observation, about the lack of counterexamples in
> testing up to large numbers:
>
> The conjecture has been checked by computer for all starting values up to 2
> 60.[15]
> <https://en.wikipedia.org/wiki/Collatz_conjecture#cite_note-Silva-15> All
> initial values tested so far eventually end in the repeating cycle (4; 2;
> 1), which has only three terms. From this lower bound on the starting
> value, a lower bound can also be obtained for the number of terms a
> repeating cycle other than (4; 2; 1) must have.[16]
> <https://en.wikipedia.org/wiki/Collatz_conjecture#cite_note-Garner-16> When
> this relationship was established in 1981, the formula gave a lower bound
> of 35,400 terms.[16]
> <https://en.wikipedia.org/wiki/Collatz_conjecture#cite_note-Garner-16>
>
> This computer evidence is not a proof that the conjecture is true. As shown
> in the cases of the Pólya conjecture
> <https://en.wikipedia.org/wiki/P%C3%B3lya_conjecture>, the Mertens
> conjecture <https://en.wikipedia.org/wiki/Mertens_conjecture> and the Skewes'
> number <https://en.wikipedia.org/wiki/Skewes%27_number>, sometimes a
> conjecture's only counterexamples
> <https://en.wikipedia.org/wiki/Counterexamples> are found when using very
> large numbers.
>
>
> Art
>
>
> On Wed, Oct 19, 2016 at 2:08 PM, Dave Philipsen <dave at davebiz.com> wrote:
>
>> Hugo, it is very interesting to me that it is so difficult to prove yet
>> anyone with a little understanding knows that it is true. It just goes to
>> show you that scientific proof is not always what it's cracked up to be.
>> You could easily write a computer program that would test all positive
>> integers up to the limitation of the size of the integer by the computer
>> and I'm sure it's already been done. And I'm sure that for as long as this
>> problem has been around someone has tested it up to some pretty large
>> numbers without finding a contradiction.  So we 'know' the conjecture to be
>> true yet we cannot prove it.
>>
>> The discussion could easily become a religious one since there are many
>> concepts which can not be scientifically proven but certain people know
>> them to be true.  As human beings we seek to prove or disprove things
>> according to our nature.  But when things are outside of our nature things
>> start to get crazy!
>>
>> Dave
>>
>>
>> On 10/19/2016 9:41 AM, Hugo Dufort wrote:
>>
>>> Collatz Conjecture
>>>
>>
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