Unsolved Problems
The long stairway problem
If one starts the maze at the prime 1583, one finds a very pecular
formation in the maze--a stairway that climbs up five levels:
1583 = 11000101111
3631 = 111000101111
7727 = 1111000101111
15919 = 11111000101111
32303 = 111111000101111
65071 =1111111000101111
But alas, this stairway is inaccessible from the main maze, for these primes
have the wrong parity. There is a similar stairway starting from the prime
33331, which by an amazing coincidence is part of the decimal stairway:
31
331
3331
33331
333331
3333331
33333331
The longest stairway within the main part of the maze starts at 1346357,
and climbs up seven levels:
1346357 = 101001000101100110101
3443509 = 1101001000101100110101
7637813 = 11101001000101100110101
16026421 = 111101001000101100110101
32803637 = 1111101001000101100110101
66358069 = 11111101001000101100110101
133466933 = 111111101001000101100110101
267684661 = 1111111101001000101100110101
Is there a longer stairway, either within the main maze or elsewhere?
Is Room 35759 connected to the main maze in some way?
This question actually has two parts: can Room 35759 be accessed from Room 2,
and can Room 2 be accessed from room 35759? This is the smallest prime for
which both questions are open. If one started at Room 35759, one can wander
though an infinite maze, reaching higher and higher primes. This "secondary
maze" seems to increase geometrically, as the main maze does. However, there
is yet to be any overlap. There may in fact be some property of the primes
connected to 35759 that prevents them from connecting to the main part of the
maze, similar to the way the primes with the wrong parity are disconnected.
But no such property has been discovered.
It is known that the primes 683, 2699, 2729, 2731, 6827, 8363, 8747, 8867,
10427, 10667, 10799, 10859, 10883, 10889, 10891, 10937, 10939, 10979, 10987,
11003, 11171, 11177, 11243, 11939, 12011, 12203, 14891, 15017, and 15083
can all be reached from 35759, and these 29 primes are precisely the primes
of correct parity less than 16384 which are apparently not in the main maze.