Prime-index-primes and twin primes

[cino hilliard]

H fans,
Consecutive prime-index-primes (PIPS)  ie., prime(prime(prime..(prime(x))..) 
and
                                                            
prime(prime(prime..(prime(x+1))..)
contain twin prime pairs.

But  prime(prime(x))  and prime(prime(prime(x))) (PIPS order 1 and 2) do not 
for some x, x+1.
Now prime(prime(prime(prime(x)))) or PIPS of order 3 contain a twinprime 
pair for all x < 33176.
This is the limit for my 2 gig ram and pari primelimit. I am guessing 
consecutive PIPS order 4,5,..n will
always contain a twin prime pair.

It seems there should be a way to tie PIPS and twin primes together. If we 
could prove PIPS of order
n must always contain a twin prime pair then we will  have a proof of the 
infinity of twin primes since
PIPS_n(x) are infinite in lieu of primes are infinite.


This Pari script is for PIPS of order 3. Print if no twin prime pair is 
found.
piptwins3(m,n) =
              {
               for(x=m,n,
               f=1;
               p1 = prime(prime(prime(prime(x))));
               p2 = prime(prime(prime(prime(x+1))));
               for(y=p1,p2,
           if(isprime(y) && y+2 <= p2 && isprime(y+2),f=0)
                 );
                 if(f,print1(x","))
                 )
                }

Maybe some one has software to check this for  > 33176 and for PIPS order 
5,6,.. to create
new sequences disproving my conjecture?

I have been thinking about a link between PIPS and twin primes for over 30 
years now and have only
been able to provide statistical evidence showing that the density of PIPS 
is less than the density of
twin primes.

Any thoughts are appreciated.

Cino

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