Gaps between prime-roots

[Rainer Rosenthal]

From: Neil Fernandez
> Rainer Rosenthal
> >In the course of thread "Primes..." in sci.math I proposed
> >the following:
> >
> >         1, 113, 1327, ... ???             (1)
> >
> >where a(n) = smallest natural such that
> >sqrt(P_(k+1)) - sqrt(P_k) < 1/n  for all k with P_k > a(n)
> 
> Are the terms 113 and 1327 proven?
> 

No.

That's why I asked here instead of throwing in some nonsense
into the OEIS.

I like your next sequence  
2,2,2,17,29,41,59,71,101,101,137,149,179,197,227,269,...

but the thrill with (1) above is: can it really be found?
I didn't even succeed in proving that

         sqrt(P_(k+1)) - sqrt(P_k) < 1/2

must hold for all k beyond some K, let alone that with P_K=113.
Even the famous Bertrand postulate (a theorem) gives me only

   sqrt(P_(k+1)) - sqrt(P_k) ~< sqrt(P_k)/2

or as I prefer writing:  w'-w ~< w/2, where w and w' are the
square roots of prime p and successor-prime p'.
Proof:  (w'+w)(w'-w)=p'-p < 2p, so w'-w < p/(w'+w) ~ (p/(2w)=w/2.

(Pun intended: p-prime is not prime p but the next one).

Rainer Rosenthal
r.rosenthal at web.de