If I understand correctly, pi being normal (which is widely believed but not known) would mean there is no max: a string of 1’s of any particular length would appear somewhere within the digits of pi.
Why would you assume that?
Referring to finding 236592 in there somewhere:
Walt was simply responding to post #2 above, in which DCnDC says:
[QUOTE=DCnDC]
For what it’s worth, I didn’t find “236592” in there, but I’m assuming that was just a number you chose arbitrarily.
[/quote]
Yes, 236592 is too in there, and not all that far in either.
Indeed, I’m aware that the OP is asking for a non-bruteforce method; I just thought that since the consensus seems to be that no such method is known or likely to exist, some people in this thread might find that site interesting…
I wonder - what’s the longest string that has the same value as the position of its starting digit? So far I got “5”. Hmm…
So you’d need the string “143” to start in the 143rd place?
90 two-digit strings, so a 1 - 0.99^90 = 0.59527 chance of there being at least one.
900 three-digit strings, so a 1 - 0.999^900 = 0.59631 chance of there being at least one.
etc.
Apparently converging to 1 - exp(-0.9)
That’s using Octave, and I don’t know if it’s handling the rounding off correctly, but if it is, it looks like about a 59% chance for each string length.
That’s ignoring strings beginning with a leading 0. Including those, it goes up to 63% chance for each string length.
There is no evidence that pi is fractal. In fact, quite the contrary - if it were fractal it would not be an irrational number, as there would be a pattern to it.
The premise of the post linked to in the OP is the same premise behind the old notion that if you give an infinite number of monkeys an infinite number of typewriters one of them will eventually write Hamlet, or War and Peace, or type out the meaning of life. It seems right but it’s not, and it’s just a product of the human mind’s inability to truly grasp the concept of infinity. Pi is infinitely long (this can be proven).
A) What do you mean by “fractal”?
B) There’s nothing preventing digit-sequences with patterns to them from representing irrational numbers; they just can’t be of the particular form of eventually-cyclic patterns.
E.g., as noted above, numbers such as 0.012345678910111213141516… or 0.101001000100001000001… have a very clear pattern to them, but are irrational all the same.
What’s that got to do with anything? Of course pi’s decimal sequence is “infinitely long” (i.e., does not eventually become constantly 0); that’s part of what fuels the OP’s hope that every particular finite decimal string appears within it.
Why do you say “It seems right but it’s not”? It’s certainly plausible that every finite decimal string appears in the expansion of pi somewhere; indeed, it’s never been disproven and is near-universally believed to be true, even by mathematicians (though, granted, it’s never actually been proven either).
By fractal I mean it has an internal grand scheme of underlying patterns, something an irrational number does not have. Picking a random string of 15 digits out of said irrational number and saying it has a pattern proves nothing.
And I do not believe an infinite number of chimps (or an infinitely long, random string of numbers) would ever produce/contain the words to the novel War and Peace (consider it a specific string of 20 million numbers, or however many individual characters are in that novel, including numbers, punctuation and spaces. It would take an infinite amount of time for that to happen, thus it would never happen. Not after a billion years, not after a billion billion years. But that’s another debate, probably for ‘great debates’ I’m just saying the post cited by the OP is derivative of this question. The infinitely long string of pi can be the combined output of the infinite number of monkeys, etc.
Patterns (or their absence) are not part of the definition of what makes an irrational number. Irrational numbers are just those that cannot be represented by simple fractions, for example, 22/7 is an approximation of pi, but 22/7 is a rational number (a ratio), whereas pi isn’t.
The monkeys/typewriters thing is a finite task - just a very big one.
Two comments:
[ol]
[li]Don’t use the word fractal that way, because no one will know what the hell you’re talking about.[/li][li]Irrational numbers can have patterns, just not repeating decimal expansions. For instance, any irrational number which satisfies a quadratic equation with integer coefficients has a periodic continued fraction expansion.[/li][/ol]
Pi is irrational. “An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers, with b non-zero, and is therefore not a rational number.” Irrational has nothing to do with internal patterns. It means only that it cannot be expressed by a terminating or repeated decimal.
This is not debatable. It’s pure math. If the expression of pi exists as generally understood, if not proven, today then it will contain any specific string of 20 million numbers.* It will do so at a finite place in the expansion. That may be very far out but not infinitely so.
In fact, there’s no such thing as infinitely far out. Infinite means endless. Any place in an expansion you can define is finite, and can be reached in finite time and expressed in a finite amount of symbols. That’s definitional, the way irrational is definitional.
- More than that. You can find any definite string of 20 million number repeated 20 million times in a row and also find it repeated 20 million times followed by a 9 and then repeated 20 million more times. And anything else you can define. Yet all of these will be found a finite number of digits into the expansion. Infinity is mind-blowing and yet well-defined.
FWIW I meant to say pi is a transcendental number rather (or in addition to) an irrational one.
Being a transcendental number does not rule out having a pattern to one’s digits. E.g., Champernowne’s constant, 0.123456789101112131415…, whose digits follow quite a simple pattern indeed, is transcendental.
(In case you are not aware of the actual definition of “transcendental”, it is this: a number is called “algebraic” if it is a root of some nonzero polynomial with integer coefficients. And “transcendental” is a fancy word for “non-algebraic”.)