Question

So, a while back I read a joke that went something like this:

“Never compute pi in binary – because it goes on infinitely and is random, it theoretically contains every finite bit string. So, you will then possess all copyrighted material in existence and be liable for some serious fines.”

This is obviously meant to be humorous, but it got me thinking. If every finite bit string exists in a binary representation of pi, would it be possible to use this as a method of transmitting data?

For example, let’s say I wanted to transmit a bit string that could be interpreted as an jpeg image. Instead of sending the information directly, I would find its location within the digits of pi, and simply send the location of the first bit within the digits of pi, as well as the lengths of the string.

This seems pretty straightforward to me, but the obvious hurtle here is that the probability of finding this string within even the first several trillion digits is remarkably small. So, it could end up taking an immense amount of time to find.

My thinking is that several machines could be dedicated to searching for large files within pi, and then creating an index of all of their start locations. So, each computation would only need to occur once and then that information could be transmitted extremely quickly from then on.

So, what do you think? Is this at all feasible, or would these computations take far too much time?

Thanks for reading! I apologize if I have overlooked any posting guidelines, this if my first question in this forum.

EDIT:

Thanks for your quick responses, folks! I figured there was error in my reasoning, nice to know why!

Answer 1

Expanding on my comments. There’s a very important concept here that’s called information entropy.

Out of full disclosure, I’m the current world record holder of the digits of Pi at 10 trillion digits (10^13).

I have approximately 10,000 copies of everyone’s social security number.

However that doesn’t mean I can just hack into everyone’s accounts and steal their identities. Because I don’t know where each person’s SSN starts. And for a typical 9-digit SSN, the first digit in Pi where that SSN will appear will be on the order of 9 digits long. In other words, the information about the SSN is kept in the address rather than in Pi itself.

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For example, if someone has the SSN: 938-93-3556

It starts at offset 597,507,393 in Pi. That number 597,507,393 is about as long as the SSN itself. In other words, we’ve gained nothing by using Pi.

_{(I’m not sure if there’s an earlier offset where it appears, but the probability decreases exponentially with smaller offsets.)}

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To generalize this, even if you had infinite digits of Pi (which theoretically holds all possible information), the address that holds data XXX will (with extreme probability) be as large as XXX itself.

In other words, the information is not held in the digits of Pi itself, but rather the address where the information starts.