20050911, 19:50  #1 
Bemusing Prompter
"Danny"
Dec 2002
California
2×17×71 Posts 
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 10^{5,000,000} 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 What do you think? 
20050911, 20:02  #2 
Jul 2003
Thuringia; Germany
2·29 Posts 
Hi ixfd64!
If is random (which is unproven, but most people think so), then you can find every finite string (of numbers) in the decimal expansion of . cyrix 
20050911, 21:35  #3 
Jun 2003
2^{4}×3^{2}×11 Posts 
As for prime numbers you can generate a prime with a large number of 11111's.
Citrix 
20050911, 23:46  #4  
Dec 2003
Hopefully Near M48
1758_{10} Posts 
Quote:


20050912, 01:04  #5 
Oct 2004
23^{2} Posts 
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 
20050912, 01:49  #6  
Aug 2002
2^{6}×5 Posts 
Quote:


20050912, 22:35  #7 
Jul 2004
Potsdam, Germany
3·277 Posts 
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? 
20050913, 02:54  #8  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
The Thue Constant (http://mathworld.wolfram.com/ThueConstant.html) is transcendental, but you'll never find two or more consecutive 0s in its base2 representation! Every 0 digit in its base2 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 basen digit strings in the constant's basen 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 20050913 at 03:01 

20050913, 03:42  #9 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
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 20050913 at 03:47 
20050913, 07:38  #10 
Bemusing Prompter
"Danny"
Dec 2002
California
2·17·71 Posts 
Yes, it's possible for transcendental numbers to have patterns.
For example, 0.1010010001000010000010000001... is one. 
20050913, 11:10  #11  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}·3·641 Posts 
Quote:
( 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) coinvented spreadspectrum radio transmission after she escaped to the U.S. from preWWII 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 20050913 at 11:24 

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