Leading digits of incomputable large numbers

In summary, the conversation was about the discovery of Graham's number and the interest in determining the first digits of incomputably large numbers. It was mentioned that the last digits of Graham's number can be determined using elementary number theory, and that leading digits of numbers not a power of ten follow Benford's Law. It was also discussed that while the first digit of e^1000000 can be easily calculated, it is not possible to determine the first digit of 3^^^3. Lastly, it was clarified that while Graham's number is not incomputable, it would still take a very long time to compute.
  • #1
BWV
1,455
1,769
Was watching a youtube on Grahams number last night and its discoverer Ronald Graham. He talked about how many algorithms can calculate the ending digits of the number (its a power of 3) but the first digit is unknown. Guessing this is generally true of all incomputably large numbers that are not powers of 10? Anyway to know the first digit of say e^1,000,000 or 10^10! or 3^^^3?
 
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  • #2
Why does anybody want to know? This is of similar interest as the 117th digit of ##\pi.##
 
  • #3
fresh_42 said:
Why does anybody want to know? This is of similar interest as the 117th digit of ##\pi.##
Ronald Graham, for one, wanted to know, it may not be of interest to know the trillionth digit of pi, but an algorithm to calculate it that was not just brute force would be interesting. But if you arent interested in the topic, don't waste your time replying
 
Last edited:
  • #4
In attempting to do this problem, one might learn new math techniques that could be applied to future problems.
 
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  • #5
For the curious with OCD tendencies, the 117th digit is: ...0

from this list of digits and excluding the decimal point:

3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230...

https://www.piday.org/million/

(I pasted the string into a text editor, deleted the decimal point and went to column 117)
 
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  • #6
jedishrfu said:
In attempting to do this problem, one might learn new math techniques that could be applied to future problems.
This is obviously the case for the last digits:
Wikipedia said:
Despite its unimaginable size, the last digits of Graham's number ##G_{64}## can be determined using elementary number theory. The last 500 digits of Graham's number are:
02425950695064738395657479136519351798334535362521
43003540126026771622672160419810652263169355188780
38814483140652526168785095552646051071172000997092
91249544378887496062882911725063001303622934916080
25459461494578871427832350829242102091825896753560
43086993801689249889268099510169055919951195027887
17830837018340236474548882222161573228010132974509
27344594504343300901096928025352751833289884461508
94042482650181938515625357963996189939679054966380
03222348723967018485186439059104575627262464195387

It can be shown that in the concatenated arrow notation Graham's number falls between ##3\rightarrow 3\rightarrow 64\rightarrow 2## and ##3\rightarrow 3\rightarrow 65\rightarrow 2##.

It is apparently not possible to cut down such numbers from behind.

To me, they are all ##O(1)##.
 
  • #7
Leading digits of numbers not a power of ten do follow Benfords Law

where the leading digit d (d ∈ {1, ..., 9}) occurs with prob
1656533102958.png


so:
1​
0.301​
2​
0.176​
3​
0.125​
4​
0.097​
5​
0.079​
6​
0.067​
7​
0.058​
8​
0.051​
9​
0.046​
 
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  • #8
 
  • #9
##e^{1000000} = 10^{1000000/\ln(10)} \approx 3.033215396802087545086402141418114327... × 10^{434294}##
For the first digit it's sufficient to calculate 1000000/ln(10) with a precision of around 0.1, which is trivial for a computer, you need about 7 digits. WolframAlpha directly delivered a few extra digits of the result.
If you ask the same question about ##{{{e^{10}}^{10}}^{10}}^{10}## then we don't know because we don't have 10^10^10 digits of ln(10) (we do have 10^10 digits however).
 
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  • #10
Computability has a specific meaning in mathematics: Graham's number is not incomputable.
 
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  • #11
BWV said:
Was watching a youtube on Grahams number last night and its discoverer Ronald Graham. He talked about how many algorithms can calculate the ending digits of the number (its a power of 3) but the first digit is unknown. Guessing this is generally true of all incomputably large numbers that are not powers of 10? Anyway to know the first digit of say e^1,000,000 or 10^10! or 3^^^3?
e^10000000 is easy. (10^10)! can still be computed completely, and you can easily find the first digits with Stirlings approximation. 3^^^3 is hopeless. you'll need to iterate fn+1 = 3fn about 7.6*10^12 times. To compute the leading digit of fn+1 you need all the digits of fn.
 
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  • #12
pbuk said:
Computability has a specific meaning in mathematics: Graham's number is not incomputable.
No, it is not, but I wouldn't wait for the TM to stop either. :wink:
 

What are "Leading digits of incomputable large numbers"?

"Leading digits of incomputable large numbers" refers to the first few digits of a very large number that is impossible to compute or calculate due to its size. These leading digits can provide insights or patterns about the number, even if the full number cannot be determined.

Why is it important to study the leading digits of incomputable large numbers?

Studying the leading digits of incomputable large numbers can help us understand the distribution and behavior of these numbers, which can have implications in fields such as mathematics, physics, and computer science. It can also provide insights into the nature of randomness and complexity in our universe.

How are the leading digits of incomputable large numbers determined?

The leading digits of incomputable large numbers are determined through various mathematical and statistical methods, such as Benford's Law, which states that the first digit of many real-world data sets follows a specific distribution pattern. Other methods may involve using logarithms or other mathematical functions to analyze the leading digits of a number.

Can we ever accurately determine the full value of an incomputable large number?

No, it is impossible to accurately determine the full value of an incomputable large number due to its sheer size and complexity. However, we can use various techniques and algorithms to approximate or estimate the value of these numbers.

What are some real-world applications of studying the leading digits of incomputable large numbers?

Studying the leading digits of incomputable large numbers has applications in various fields such as finance, fraud detection, and data analysis. It can also be used to test the validity of data sets and identify potential errors or anomalies in the data. Additionally, understanding the distribution of leading digits can help in designing more efficient algorithms and data compression methods.

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