----- Original Message ----- From: "N. J. A. Sloane" <njas@research.att.com> To: <math-fun@mailman.xmission.com> Cc: <njas@research.att.com> Sent: Monday, January 08, 2007 9:15 AM Subject: [math-fun] Re: Car Talk and prime numbers
I asked this list on Saturday about deletable primes. Thanks to everyone who replied. But no one answered my specific question about the largest known deletable prime, so I'm adding these two sequences to the OEIS. Maybe someone would like to extend them! (I will also update them with references to various links that people sent me.) Neil
A096243 shows that (number of deletable primes with d+1 digits)/(number of deletable primes with d digits) grows fairly steadily with increasing d. It is not unreasonable to conjecture that this value approaches 10, just as does (number of primes with d+1 digits)/(number of primes with d digits). In other words, I think it is reasonable to conjecture that the deletable primes are infinite in number. I imagine that a statistical case could be made that most primes are insertable, that is, some digit can be inserted to produce a larger prime. At any rate, I was only able to find 61 non-insertable primes <= 10^8. This means that large deletable primes should be fairly easy to generate, as Phil Carmody demonstrated by generating a (probable?) 68-digit deletable prime in less than a second of computer time. I believe the answer to your specific question about the largest known deleteable prime was the 300-digit prime given at http://primes.utm.edu/curios/page.php?number_id=561&submitter=Jobling I don't know if this number has been proved to be a deleteable prime, but I doubt anyone has bothered to try to find a larger deletable prime. I think the main impediment to finding such primes would be the primality testing.