Title says it all.
Every day, a random number between 0 and 65535 is generated by the game. If this number matches the last two bytes of the personality value of any of the Pokémon in the party (that is, the remainder when their personality value is divided by 65536), Mirage Island will appear for that day. The number is generated at 0:00 hours every day and it is active for 24 hours. Even though play time has nothing to do with the real-time clock, a game that has reached the maximum time of play (999 hours and 59 minutes) will not generate Mirage Island numbers anymore. Instead, the Mirage Island number will always be zero. However, if the game's real-time clock is not functioning (the internal battery has died) at the moment when time of play reaches 999 hours 59 minutes, then the Mirage Island number will permanently stay at what value it was before. - Bulbapedia
