Can anyone give a heuristic argument (based on the density of the primes and their approximate independence from one another) that the property ought to hold for all sufficiently large n? Jim Propp On Thu, Apr 30, 2015 at 10:12 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
I conjecture it's true for all n (and verified for 3, 4 as well). Here's an app which allows you to see the graph of primes connected by single-digit replacements:
Manipulate[ GraphPlot3D[ Flatten[Map[ Function[{n}, Map[(n -> #) &, Select[Flatten[ MapIndexed[ Table[n + (i - #) 10^(#2[[1]] - 1), {i, # + 1, 9}] &, Reverse[IntegerDigits[n, 10]]]], PrimeQ]]], Select[Table[i, {i, 10^(digits - 1), 10^digits}], PrimeQ]]], VertexLabeling -> True], {{digits, 2}, 1, 4, 1}]
Sincerely,
Adam P. Goucher
Sent: Thursday, April 30, 2015 at 2:17 PM From: "James Propp" <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] Prime ladders
For what values of n is it possible to get from every n-digit prime number to every other by way of a succession of single-digit alterations?
It's trivially true for n=1, and it's also true for n=2 since every 2-digit prime remains prime if you change its first digit to a 1.
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