First, what about other bases besides ten? [This is normally dan asimov's question, isn't it?] Second, the probability calculation is ignoring the fact that there are several ways to add a digit and get the same result -- in particular, inserting a digit next to an identical digit produces the same number no matter which side of the digit the insertion happens on. This immediately reduces your "10n" to at most "9.5n", I think. The reduction is relatively more severe for base 2. --ms -----Original Message----- From: math-fun-admin@mailman.xmission.com [mailto:math-fun-admin@mailman.xmission.com]On Behalf Of Michael Kleber Sent: Thursday, February 27, 2003 12:08 To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Prime Question: Whoops No no no, these probabilistic arguments are going the wrong direction.
Let a prime be a good prime if a digit can be deleted leaving either the empty string or a good prime. Are there infinitely many good primes?
Let's be very naive and say that all we know is the prime number theorem, Pr(n is prime) =~ 1/ln(n). Then E(# n-digit good primes) / E(# (n-1)-digit good primes) =~ (# ways to add a digit) * (Pr(an n-digit number is prime), which is around (10n) * (1/(n ln 10)) = 10/ln 10 =~ 4.34. So the number of good primes should grow exponentially. In fact, the quick Mma hack gives this count (NJAS note): # n-digit good primes: 4, 16, 94, 585, 3788, 25768, 182762 Successive ratios are 4., 5.875, 6.2234, 6.47521, 6.80253, 7.0926, so 4.34 looks like a serious underestimate. I'd guess most of the difference comes from the higher odds of getting a prime after adding a digit 0,3,6,9 for congruence mod 3 reasons that have already been mentioned. Hmm: if this goes in the EIS, let's use the apparently accepted term, "deletable prime" (says Google). --Michael Kleber kleber@brandeis.edu _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun