919 conjecture

[franktaw at netscape.net]

No.

It is an open problem of long standing to show that there is always a 
prime between n^2 and (n+1)^2.  This means that we can't prove that 
prime gaps are always less than sqrt(n) for n sufficiently large.

Franklin T. Adams-Watters

-----Original Message-----
From: Tanya Khovanova <tanyakh at TanyaKhovanova.com>

Hello all,

I was looking at A069803 - Smaller of two consecutive palindromic 
primes:   2,
3, 5, 7, 181, 787, 919
Conjectured to be complete.

I am interested in seeing a proof that 919 is actually the largest 
palindromic
prime such that the next prime is palindromic.
I checked up to 10^8 with Mathematica coding.

Also, it is obvious that the distance from a palindrome n to the next 
one is
more than Sqrt(n/10). It is clear that prime gaps grow slower than 
that. Looking
at the prime gaps sequence A053303, it is easy to prove that 919 is the 
last
number like that up to 10^16.

Is there a bound for prime gaps that proves that the gaps are less than
Sqrt(n/10) starting from some n?

Tanya



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