The Digits of Pi Cannot Be Sequentially Generated By A Computer Program

2007.11.26   prev     next

Question: Can you write a program on your current computer, to generate digits of pi indefinitely?

Don’t worry about how it will output them — perhaps it will flash them up on your monitor one at a time. And don’t worry about your computer breaking, or running out of power, or our universe coming to an end, or anything like that. Assume the computer you’re using right now will run fine forever.

Can you write a program to sequentially display the digits of pi that has no programmatic failure date?

Answer: No. You can’t.

Why not? Because all the bits in your computer — including various state bits inside the processor itself — as a group, have a finite number of unique combinations. So no matter what program your computer is running, it must eventually start repeating itself. Since the digits of pi are non-repeating, no program on your computer can, even theoretically, generate them correctly.

This is also true of any non-repeating sequence, even if it isn’t remotely random-looking. For example, can you write a program that will display this sequence of digits and (theoretically) never fail:

010010001000010000010000001...etc.

Answer: No. You can’t.