Prev: Compactness in metric spaces
Next: Applying Cauchy formula...
From: OwlHoot on 8 Jul 2010 01:07
On Jul 7, 7:04 pm, OwlHoot <ravensd...(a)googlemail.com> wrote:
> On Jul 7, 4:51 pm, Ludovicus <luir...(a)yahoo.com> wrote:
>
> > Conjecture:
> > If a sequence Un is such that LogLog(Un) is of the order Log(n) or
> > less
> > and it contains five or more primes, then the sequence will contains
> > infinitely many primes. (No counting in the five, the numbers used to
> > initiate the algorithm.)
> > This conjecture will comprise many of the unsolved prime conjectures:
> > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc
> > But not Fermat primes.
> > Ludovicus
>
> Couldn't you just define the sequence to be the first 10 primes,
> and the primes plus 1 thereafter? (assuming the infinite sequence
> of primes has the property)

To clarify, this was intended to be a counterexample, because
by definition it doesn't contain an infinite number of primes.


From: Tim Little on 8 Jul 2010 04:52
On 2010-07-07, Ludovicus <luiroto(a)yahoo.com> wrote:
> Yes . I forget to stablish that the sequence must be infinite and
> produced by the uniform application of a given algorithm.

Starting with 2^17-1, iterate U_(n+1) = floor(U_n/2). This infinite
sequence is produced by the uniform application of a given algorithm,
contains 5 primes after the "initialization" value, and no more.


- Tim
From: jbriggs444 on 8 Jul 2010 08:15
On Jul 7, 1:08 pm, Ludovicus <luir...(a)yahoo.com> wrote:
> On 7 jul, 12:38, Ross <rmill...(a)pacbell.net> wrote:
>
>
>
>
>
> > On Jul 7, 8:51 am, Ludovicus <luir...(a)yahoo.com> wrote:
>
> > > Conjecture:
> > > If a sequence Un is such that LogLog(Un) is of the order Log(n) or
> > > less
> > > and it contains five or more primes, then the sequence will contains
> > > infinitely many primes. (No counting in the five, the numbers used to
> > > initiate the algorithm.)
> > > This conjecture will comprise many of the unsolved prime conjectures:
> > > Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc
> > > But not Fermat primes.
> > > Ludovicus
>
> > You need to define what sequences you are considering.  How about:
> > Un=2,3,5,7,11 for n=1 to 5
> > Un=10^n for n>=6?
>
> Yes . I forget to stablish that the sequence must be infinite and
> produced
> by the uniform application of a given algorithm.
>
> Your example do not conform with the conditions because your five
> numbers
> are precisely, the given for initiate the algorithm.
> Ludovicus- Hide quoted text -
>
> - Show quoted text -

Now all you need to do is to rigorously define what it means for
a particular number to be "given for initiate the algorithm".

Can you, for example, provide a procedure that we can use to
examine an algorithm to determine whether it "contains five
prime numbers" as "initial conditions".

What if an algorithm uses "1+1" instead of 2?
Or (courtesy of Tim Little) 2^17-1 instead of 131071?

What if an algorithm contains a primality check, examines
the output it would otherwise generate and refuses to
generate any prime after the fifth?

What if an algorithm generates any primes at all? In what
sense are any of those generated prime numbers _not_
implicitly contained in the algorithm?
From: Ludovicus on 8 Jul 2010 10:32
On 7 jul, 11:51, Ludovicus <luir...(a)yahoo.com> wrote:
> Conjecture:
> If a sequence Un is such that LogLog(Un) is of the order Log(n) or
> less
> and it contains five or more primes, then the sequence will contains
> infinitely many primes. (No counting in the five, the numbers used to
> initiate the algorithm.)
> This conjecture will comprise many of the unsolved prime conjectures:
> Twin primes, Fibonacci primes, Polynomial primes, Mersenne primes, etc
> But not Fermat primes.
> Ludovicus

New wording of Ludovicus Conjecture:
"If by the uniform application of an algorithm, an infinite integer
sequence Un results, such that LogLog(Un) is of the order of Log(n)
or less, and it contains five or more primes of certain type,
(Not counting the numbers given in the definition of the algorithm)
then it will contains infinitely many primes of that type.
It is supposed that the functions floor or ceiling are not used."

Examples: In twin primes , Fibonacci,Mersenne,and polynomial
sequences the two conditions are fulfiled.
But not in Fermat's sequence because LogLog(Fn) = 2^n + Log2.
Nor in Un = 10^n + 1 because only 11 and 101 are known as primes.
Ludovicus
From: Tim Little on 8 Jul 2010 11:03
On 2010-07-08, Ludovicus <luiroto(a)yahoo.com> wrote:
> New wording of Ludovicus Conjecture:
[...]
> It is supposed that the functions floor or ceiling are not used."

Heh, a classic application of refining a conjecture by ruling out very
specific inconvenient counterexamples without thinking about the
principles illustrated by those counterexamples.


- Tim
First  |  Prev  |  Next  |  Last
Pages: 1 2 3
Prev: Compactness in metric spaces
Next: Applying Cauchy formula...