There is a special place in the netherworld reserved for mathematicians who play with prime numbers. It consists of a corridor 1000 miles long lined with 1,000,000 doors. Each of the 6666 souls imprisoned there is assigned a prime number, 2, 3, 5, 7, 11, etc.

At the start of the first month all of the doors are closed. Each person must walk down down the corridor, opening or shutting every P-th door, according to the given prime. For example, the soul assigned #5 opens or shuts doors 5, 10, 15, 20, etc.

At the end of a month they have all reached the end of the corridor, and they must start back, performing the operation in the opposite order. For example the #7 soul opens or closes doors 999994, 999987, 999980, 999973, and so forth.

How many months will it be before the doors are all closed again?