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Old 2005-09-11, 19:50   #1
ixfd64
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Default mathematical paradox?

Here is a problem I have been wondering about for quite some time. Is it possible for transcendental constants (like pi or e), or perhaps very large prime numbers (like those over 105,000,000 digits), to have long repeating strings of digits or other interesting patterns?

In other words, is it possible for pi to have, say, a string of five trillion ones?

Code:
11111.....(4,999,999,999,990 ones).....11111
The chances of this is one out of this is nearly zero. But then, pi goes on infinitely.

What do you think?
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Old 2005-09-11, 20:02   #2
cyrix
 
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Hi ixfd64!

If  \pi is random (which is unproven, but most people think so), then you can find every finite string (of numbers) in the decimal expansion of  \pi .

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Old 2005-09-11, 21:35   #3
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As for prime numbers you can generate a prime with a large number of 11111's.

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Old 2005-09-11, 23:46   #4
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Quote:
Originally Posted by ixfd64
The chances of this is one out of this is nearly zero. But then, pi goes on infinitely.
If the decimal expansion of \pi is random (obviously, \pi itself is not random), then such a sequence of ones must occur. (Actual) infinity wins out over "nearly zero".
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Old 2005-09-12, 01:04   #5
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Just out of interest, the BINARY expansion of EVERY MERSENNE PRIME is a long string of 1s without zeroes.

There is not yet a proof that the decimal expansion of pi (or certain other numbers) has its digits randomly arranged. If it is then it is possible to find within it any arbitrary string of digits. If not, you might find a certain string , or might not. For example there is a number whose decimal expansion contains only odd digits eg 1/3. You will not find any sequence in there looking like 88888 etc
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Old 2005-09-12, 01:49   #6
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Quote:
If Pi is random (which is unproven, but most people think so),
Pi is a constant. You mean if the digits of Pi are normally distributed.
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Old 2005-09-12, 22:35   #7
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Question

Is there a need for some form of distribution at all?

IANAM, but I'd think that it's enough that Pi is transcendent, which AFAIK means that the decimal representation

a) has an infinite amount of positions after the decimal point and
b) doesn't have a recurring decimal.

As a result, every finite pattern should be included, right?
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Old 2005-09-13, 02:54   #8
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Quote:
Originally Posted by Mystwalker
IANAM, but I'd think that it's enough that Pi is transcendent, which AFAIK means that the decimal representation

a) has an infinite amount of positions after the decimal point and
b) doesn't have a recurring decimal.

As a result, every finite pattern should be included, right?
Not necessarily. Ya gotta be careful with those transcendentals! Just because a number is transcendental doesn't mean it has every "special" property. Your a) and b) are correct, but they are not sufficient to guarantee that every finite pattern is included.

The Thue Constant (http://mathworld.wolfram.com/ThueConstant.html) is transcendental, but you'll never find two or more consecutive 0s in its base-2 representation! Every 0 digit in its base-2 representation has a 1 digit on each side, due to its definition. Now, this just means that not every finite binary pattern appears in the binary representation of this particular transcendental constant, but I'm fairly sure that for any base n, a constant can be constructed that is transcendental (proving transcendence is the hard part) but does not include certain finite base-n digit strings in the constant's base-n representation.

- - - -

Check out "We are in Digits of Pi and Live Forever" at http://sprott.physics.wisc.edu/pickover/pimatrix.html !!!

Last fiddled with by cheesehead on 2005-09-13 at 03:01
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Old 2005-09-13, 03:42   #9
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I checked the "We are in Digits of Pi" link and, being lazy, read only the first two posts. But it seems that at least the first two posters there understand neither the definition of an irrational number nor cheesehead's point.

In any case, if you want to find a number that certainly does contain every finite sequence of numbers, there's a much easier way to do so. Just consider the number:

0.123456789101112131415161718...

Here's another transcendental number that doesn't contain every possible combination of (binary) bits:

http://mathworld.wolfram.com/LiouvillesConstant.html

For instance, the sequence 1001 never appears in that number.

Last fiddled with by jinydu on 2005-09-13 at 03:47
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Old 2005-09-13, 07:38   #10
ixfd64
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Yes, it's possible for transcendental numbers to have patterns.

For example, 0.1010010001000010000010000001... is one.
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Old 2005-09-13, 11:10   #11
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Quote:
Originally Posted by jinydu
I checked the "We are in Digits of Pi" link and, being lazy, read only the first two posts. But it seems that at least the first two posters there understand neither the definition of an irrational number nor cheesehead's point.
It gets better as it goes along, like many of our threads. Please read at least down to the Britney Spears photo. And the second page is better (not counting photo).

( Why Britney Spears? Well, it may be because of the "Britney Spears guide to Semiconductor Physics: semiconductor physics, Edge Emitting Lasers and VCSELs" page at http://britneyspears.ac/lasers.htm Or not.

But be sure to read how actress Hedy Lamarr (a genuine electrical engineer) co-invented spread-spectrum radio transmission after she escaped to the U.S. from pre-WWII Austria. U.S. Patent 2,292,387 for the "Secret Communication System" was granted on August 11, 1942. The patent is actually under her married name at the time - Hedy Kiesler Markey.

http://britneyspears.ac/physics/intro/hedy.htm)

Last fiddled with by cheesehead on 2005-09-13 at 11:24
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