paper published

[mensa...@aol.com]

Johan van der Galien wrote:
> Dear Newsgroup Readers,
>
> Paul Stadfeld has submitted a paper to the open access scientific
> internet journal: Scientia Araneae Totius Orbis (S.A.T.O. for short).
> It is peer reviewed by myself and two other authors of S.A.T.O.
> publications. The paper is submitted, revised, accepted and published
> on the S.A.T.O. site.
>
> The abstract is given below:
>
> Title: Blueprint for Failure: How to Construct a Counterexample to the
> Collatz Conjecture
> Author: Paul Stadfeld
>
> Abstract:
> A proof of the Collatz Conjecture has so far been elusive. But it may
> easily be resolved by finding a counterexample. In this paper we study
> the 3n+C extensions of the Collatz Conjecture (most of which fail) to
> learn what a counterexample to 3n+1 would look like if it exists.
> The study is based on the novel approach of creating a data structure
> (Sequence Vector) based on relative Collatz sequences without regard to
> their values. Functions are then derived from the data structure to
> allow algebraic analysis of the Collatz sequences.
> These functions (Hailstone and Crossover Point) are the key to
> understanding why 3n+1 works and why 3n+5 fails. This understanding
> will then permit the construction of counterexamples as opposed to
> simple brute force searching. Furthermore, we will find that
> counterexamples must meet stringent structural requirements (the
> Blueprint) that will tell us when a counterexample CANNOT be
> constructed, which has deep implications concerning proving the Collatz
> Conjecture.
> We will also gain new insight on the relationship between the positive
> and negative domain with regard to 3n+1, that they are not separate
> problems but are actually a single problem from the viewpoint of
> relative Collatz sequences. This, too, has deep implications, for
> certain proofs in one domain imply proof in the other which seems to
> have been overlooked up to now with startling consequences.
> The Collatz Conjecture will NOT be proved or disproved in this paper.
> What will be shown are tools and insights that may further the search
> for an actual proof.
>
> Full paper at S.A.T.O. Volume 5.3. (2006) http://home.zonnet.nl/galien8
>
> Also the author has questions for you readers (The author will engage
> in the discusion)
>
> The Collatz conjecture has been tested for hundreds of
> quadrillions of numbers. But that's only 18 digit numbers.
> If the quantity of numbers checked doubled every year,
> it would take 2 3/4 centuries to reach 100 digit numbers.
> But there are many sets of numbers with interesting Collatz
> properties whose smallest member has hundreds or thousands
> of digits. What if the set of counterexamples also had such
> huge numbers?
>
> They would never be discovered. But the interesting numbers
> weren't "discovered', they were constructed. Construction
> allows us to find examples that can never be discovered by
> brute force searching of all possible intergers. Is it possible
> that counterexamples can be constructed?
>
> That would require knowing the structure of a counterexample.
> (With structure is actually meant the sequence vector as described in
> the paper.)
> But how do we know what the structure is if there are no actual
> counterexamples to study?
>
> And suppose we had the structure and still couldn't construct
> a counterexample. Would that prove the conjecture true?
>
> Please answer the questions.
>
> Kind regards,
>
> Johan G. van der Galien.
> Scientia Araneae Totius Orbis
> Chief-Editor and Webmaster