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.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun