919 conjecture

[David Wilson]

The graph of A002375 strongly indicates that the number of Goldbach 
partitions of even n >= 4 remains positive. Similarly, the graph of A014085 
strongly indicates that the number of primes between adjacent positive 
squares remains positive. Though neither conjecture is proved, and a freak 
departure from the visible pattern is theoretically possible, the trends in 
the empirical evidence suggest that both conjectures are a very safe bet. 
Conway suggests calling such conjectures, with ample empirical evidence and 
little reason to doubt that evidence, "sureties". Thus Goldbach's and 
Legendre's conjectures are "sure" if not provably true.

I suggest that you count and graph the number of primes strictly between 
adjacent palindromes. I suspect that, except for anomalous adjacent 
palindrome pairs (10^k-1, 10^k+1) (which cannot be adjacent palindromic 
primes), the number of intervening primes will exhibit a visually convincing 
growth pattern that will convince you that (919, 929) is indeed the last 
pair of consecutive palindromes and consecutive primes.

----- Original Message ----- 
From: "Tanya Khovanova" <tanyakh at TanyaKhovanova.com>
To: <seqfan at ext.jussieu.fr>
Sent: Tuesday, July 17, 2007 6:12 PM
Subject: 919 conjecture


> 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
>
>
> _________________________________________________________________
> Need personalized email and website? Look no further. It's easy
> with Doteasy $0 Web Hosting! Learn more at www.doteasy.com
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.476 / Virus Database: 269.10.8/906 - Release Date: 7/17/2007 
> 6:30 PM
>