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

[Arthur Flexser]

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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