I have to imagine I’ve run across this before, but here’s a nice little page which allows one to search the (base ten) digits of π for any sequence of up to 120 digits that one wants. It searches the first 200 million digits of pi, which is enough digits that you can be reasonably sure that any string of six or seven digits you look for are there, and it’s not ridiculously unlikely that a string of ten digits in a row will turn up. The natural question is, why is this interesting?
People who’ve learned a bit about pi may have heard that it’s probably a “normal number”, that is, a number whose digits contain every possible finite string of digits within it somewhere. That suggests that finding any particular string of digits in pi is no more surprising than finding any particular word in a complete dictionary (if we imagine there’s a dictionary that ever did include all the words of a language). The story’s a little more complicated than that.
The first is we don’t have all the digits of pi to look through. Something like ten trillion (decimal) digits are known, and my data might be old, but pi is an irrational number, going on infinitely long without falling into a repeating loop. So the case is actually more like happening to find any particular word in a single page ripped out of a dictionary. That’s enough to make the finding, or the failure to find, a string of digits interesting.
The decimal digits of pi are mildly irritating to figure out, since there’s not (as far as I know) any way to generate digits without knowing all the ones before it. There are, oddly, formulas to generate one of the digits of pi in binary without calculating the ones before it (and therefore it’s easy to calculate the hexadecimal digits of pi). So if we want the hexadecimal digits we could set up many computers to work out their individual digits, independently of one another, and get a lot of digits just about simultaneously. But that doesn’t help with the decimal digits, as far as I’m aware, and we can’t find them by dividing up the task among multiple computers.
The next thing that makes searching in pi interesting is that we don’t actually know that pi is a normal number. It’s thought to be, on several grounds. The easiest ground to understand is that we can test how often the short sequences of digits turn up in the digits of pi we do know, and those turn up just as often as we’d expect from a normal number. This isn’t proof, but it’s suggestive.
More nearly a proof, though, is an argument that’s almost a probability one: just about all numbers are normal ones. It’d be a rare stroke of luck for pi to be an exception. It’s not technically impossible, but for right now, the state of affairs is a little bit like this: think of all the numbers — positive and negative, whole numbers, fractions, rationals, irrationals, even complex numbers if you like — and pick one. I can be reasonably certain that you did not pick the number , as there are so many numbers which are not that. My reasonable certainty isn’t proof that you didn’t pick that number, but I wouldn’t believe it if you claimed you had picked that one.
So there’s good reason to think that pi is normal, but it isn’t proven, and there’s a chance that it’s not.
Something I find irresistible about normal numbers is that while we know nearly every number is normal, we don’t have many examples of them. I believe the only numbers who are known to be normal are ones which were identified because we knew they would have to be normal. For example, in base ten, the number created by 0.123 456 789 101112 131415 161718 (et cetera) is normal, but that’s not a number anyone cares about for anything except being normal. Numbers that have some common use, like pi, or e, or the square root of two, aren’t known to be normal.
(Wolfram Mathworld mentions a paper that means to prove the square root of two is a normal number, but notes the work uses a “nonstandard approach” — I don’t know whether that means it uses an unusual approach or whether it uses one in a field called nonstandard analysis which I don’t know enough about — and so the result isn’t universally accepted. It may seem strangely experimental-science to have a mathematics paper that is accepted for publication but not necessarily believed. Difficult or exotic or novel proofs can be like that, though.)
A bit of pure fun to be had with a search-the-digits page like this is to start with some number — let’s say 2012 while it isn’t quite out of date — and see where that first turns up, which here is position number 7200 after the decimal. The number 7200 first turns up in the 12,123rd position. 12,123 first turns up in the 40,837th position. But 40,837 first turns up in the 15,083rd position, and just as we might have started suspecting that this process was going to make us plunge deeper and deeper into the digits of pi. 15,083 first turns up in the 35,803rd spot and surely the shuffling around of the 5, 0, 8, and 3 has to please some mildly neurotic, numerological impulse of people. 35,803 takes us to the 89,992nd spot; 89,992 to the 17,118th, and 17,118 to the 89,027th.
Where does this lead? The thing I would wonder is, are there numbers which turn into loops? There are at least a few: Dan Sikorski is reported as noting that 169 turns up at the 40th position, and 40 at the 70th, 70 at the 96th, and so on in a sequence that eventually comes to 9385, which turns up at the 169th position, and so on.
Some numbers collapse: 211 is first found in the 93rd position; 93 is first found in the 14th position; and 14 is first found in the 1st position. Since 1 is also found in the 1st position, the result is like falling into a pit. (Doug Hafen is credited by the Pi-Search Page with that discovery.)
What I haven’t seen, and wonder about, is whether there are reasonable bounds on a chain of digits. For example, starting from 2012, we got a sequence of positions that seemed to not get less than about 10,000 (after the first few attempts) nor more than 100,000. Would that pattern hold up if we found where 89,027 first turns up? (Yes.) Or for the number after that? (Yes.) After that? (You already looked, didn’t you?) If 10,000 is too low a bound, or 100,000 too high, are there provable bounds? Are there typical bounds?
For that matter, is there any number we could start from which would give us an ever-increasing sequence, where we just have to keep going deeper and deeper into pi to find the entries, maybe reaching one that can’t be found in the ten trillion digits we know?
Does this mean anything? I don’t know. There might be something interesting in following it. There might not. It’s, as best I know, an open question, and one that can be fun to play with and, sadly, probably pretty hard to prove. But, ah, the searching is a delight.