There exists no Hamiltonian cycle for 4-digit primes because 6983 is connected only to 6883. Similarly, there is no Hamiltonian cycle for 5-digit primes because 46769 is connected only to 96769. There is no Hamiltonian cycle for 6-digit primes because there is a 6-digit weakly prime. Warut On Thu, Mar 12, 2009 at 1:40 PM, Edwin Clark <eclark@math.usf.edu> wrote:
You can also make prime circles:
I find that the graphs of k-digit primes are not only connected, but are Hamiltonian for k = 1,2 and 3. I haven't tried k = 4 yet. --nor have I tried for bases other than 10.
Here are the Hamiltonian cycles for each of these cases:
[2, 3, 5, 7, 2]
[11, 13, 17, 19, 29, 23, 43, 53, 59, 79, 89, 83, 73, 71, 31, 37, 47,97, 67, 61, 41, 11]
[101, 103, 503, 523, 223, 823, 863, 163, 563, 593, 193, 293, 263, 269, 569, 509, 409, 809, 109, 709, 701, 401, 601, 661, 461, 761, 769, 739, 439, 839, 139, 239, 229, 227, 257, 457, 557, 587, 547, 541, 241, 941, 641, 691, 191, 991, 997, 977, 971, 911, 211, 811, 311, 331, 631, 131, 431, 491, 421, 521, 571, 271, 277, 877, 577, 677, 647, 347, 947, 967, 937, 337, 367, 307, 907, 107, 607, 617, 317, 397, 797, 757, 157, 857, 887, 883, 283, 983, 683, 613, 673, 773, 743, 733, 233, 433, 463, 443, 643, 653, 953, 353, 853, 859, 829, 929, 919, 419, 719, 619, 659, 359, 349, 149, 179, 173, 113, 313, 373, 383, 389, 379, 479, 449, 499, 599, 199, 197, 137, 167, 467, 487, 787, 727, 127, 827, 821, 881, 281, 251, 751, 151, 181, 101]
--Edwin
On Wed, 11 Mar 2009, Warut Roonguthai wrote:
I found that all the 4-digit primes are connected even under the restriction, and this is also true for the 5-digit case.
Too lazy for larger cases ...
Warut
On Tue, Mar 10, 2009 at 11:41 PM, <rcs@xmission.com> wrote:
3 -> 13: My notion was that leading 0s are ruled out, so each size is a separate problem space. With this restriction, I think all the two digit primes are connected, and all the three digit primes are connected. The average number of connections is bounded, so there should be a few islands among the larger primes.
The situation is interesting for other radices: Binary just barely hangs together for 5-7, is split for 11,13, and has a stringy graph for 17...29.
Ternary (and other odd radices) split into disconnected pieces, based on the pattern of even and odd digits.
If we allow leading 0s, I'd tend to agree with ACW, but consider one piece of contrary evidence: There's a five-digit number N such that N + 2^K is always composite.
Rich
--------------
Quoting Allan Wechsler <acwacw@gmail.com>:
Do the rules permit you to go from 3 to 13? If yes, then I conjecture you can get from any prime to any other prime.
On Mon, Mar 9, 2009 at 1:11 AM, Warut Roonguthai <warut822@gmail.com> wrote:
9973 -> 1973 -> 1913 -> 1013 -> 1019 -> 1009 9973 -> 1973 -> 1933 -> 1033 -> 1039 -> 1009 9973 -> 1973 -> 1979 -> 1949 -> 1049 -> 1009 9973 -> 9173 -> 9103 -> 1103 -> 1109 -> 1009 9973 -> 9173 -> 9103 -> 9109 -> 1109 -> 1009
Warut
On Mon, Mar 9, 2009 at 10:30 AM, <rcs@xmission.com> wrote:
The Prime Ladder puzzle is to change one prime into another, by changing a digit at a time. All the intermediate numbers must also be prime.
Example: 101 into 997: 101 -> 107 -> 197 -> 997
Challenge: Change 1009 into 9973.
Rich
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