Primes between consecutive prime-index-primes

[cino hilliard]

Hi,
Thanks Farideh,Frank for resolving some of my issues on this subject.

>From: f.firoozbakht at sci.ui.ac.ir
>CC: Frank Ellermann <nobody at xyzzy.claranet.de>
>Subject: RE: Primes between consecutive prime-index-primes

>As Frank wrote the first conjecture is equivalent to twin prime conjecture,
>because the set of primes between prime(prime(n)) and prime(prime(n+1)) is 
>:
>
>A(n)={prime(prime(n)+1),prime(prime(n)+2),...,prime(prime(n+1)-1)}
>so #A(n)= prime(n+1)-1-(prime(n)+1)+1 ,or #A(n)=prime(n+1)-prime(n)-1.

prime(n+1) - prime(n)-1 = 1 implies prime(n+1) - prime(n) = 2
Hence #A(n)= 1 for infinite n's Iff prime(n+1) and prime(n) are twin primes.
Then since there are an infinity of prime(n) and prime(n+1)'s it follows 
that
when they differ by 2 they are twin primes and if they differ by 2 an 
infinite number of
times then there is an infinite number of twin primes.

It seems to me that this is a more general statement than the twin prime 
conjecture since
we can examine the possibilities of the infinity of sole prime triples, 
quintuples etc. Eg.,
Conjecture: The primes that are  "only" 3 primes between two consecutive 
prime-index-primes
are infinite.

You disposed of the case there can be no double, quadruple,2k  such primes 
below. Obviously the
odd case can be extended to any number of terms.

Conjecture:
For all k=0,1,2,.. there is an infinity of consecutive prime-index-prime 
pairs such that exactly 2k+1 primes
are between them.

>Hence #A(n)= 1 for infinite n's Iff the twin prime conjecture is true.

Could we say Iff #A(n) = 1 for infinite n's then the twin prime conjecture 
is true?

>
>Also the second conjecture is true because #A(n)= prime(n+1)-prime(n)-1 is 
>odd
>for n > 1.

Cino

>
>
>Regards, Farideh.
>
>
>
>Quoting cino hilliard <hillcino368 at hotmail.com>:
>
> >
> > Henry in Rotherhithe wrote:
> >
> > >From: "Henry in Rotherhithe" <se16 at btinternet.com>
> > >To: "cino hilliard" <hillcino368 at hotmail.com>,
> > <seqfan at ext.jussieu.fr>
> > >Subject: RE: Primes between consecutive prime-index-primes
> > >Date: Sat, 1 Nov 2003 00:56:40 -0000
> > >
> > >It looks rather simple.
> >
> > Egads, indeed it was. Thank you and others for the explanation.
> >
> > >
> > >1) For x>1, prime(x) is odd.
> > >2) There is a even number (e.g. y = prime(x)+1) between prime(x) and
> > >prime(x+1) for x>1.
> > >3) There is a prime number (namely prime(y)) between prime(prime(x))
> > and
> > >prime(prime(x+1)) for x>1.
> >
> > I have submitted another sequence that lists the "only" 1  prime between
> >
> > prime(prime(x)) and
> > prime(prime(x+1)).
> >
> > 7,13,37,61,113,181,281,359,557,593,787...
> >
> > Conjecture: The primes that are the "only" prime between two
> > consecutive
> > prime-index-primes are infinite.
> >
> > Also sequence for only 3 primes and trajectory for 1,3,5..2k+1 primes
> >
> > Conjecture: There cannot be an even number total of primes between two
> > consecutive prime-index-primes.
> >
> > Maybe these can also be disposed of easily also.
> >
> > Cino

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