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

[Dave Philipsen]

Intriguing, but way over my head.  Hope you achieve your goal.

Dave Philipsen

> On Oct 20, 2016, at 8:57 AM, Hugo Dufort <hugo at seshat.ca> wrote:
> 
> 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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