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.