Primes between consecutive prime-index-primes

[f.firoozbakht at sci.ui.ac.ir]

Dear Seqfans,

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.

Hence #A(n)= 1 for infinite n's Iff 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.


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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