Leading digits of incomputable large numbers

  • Context: Undergrad 
  • Thread starter Thread starter BWV
  • Start date Start date
  • Tags Tags
    Large numbers Numbers
Click For Summary

Discussion Overview

The discussion centers on the leading digits of incomputable large numbers, particularly in relation to Graham's number and other large exponentials like e^1,000,000 and 10^10!. Participants explore the challenges and methods for determining these digits, as well as the implications of such inquiries.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants note that while algorithms can calculate the last digits of Graham's number, the first digit remains unknown, suggesting this might be true for all incomputable large numbers that are not powers of 10.
  • Others question the relevance of knowing the first digit of such large numbers, comparing it to the interest in specific digits of π.
  • Some participants argue that exploring these problems could lead to new mathematical techniques applicable to other areas.
  • One participant mentions that leading digits of numbers not a power of ten follow Benford's Law, providing probabilities for each leading digit from 1 to 9.
  • Another participant explains that for e^1,000,000, calculating the first digit can be done with sufficient precision using logarithmic properties, while noting the complexity increases for larger exponentials like 3^^^3.
  • There is a clarification that Graham's number is not incomputable, which leads to further discussion about the nature of computability in mathematics.

Areas of Agreement / Disagreement

Participants express differing views on the significance of determining leading digits of large numbers, with some finding it worthwhile and others questioning its relevance. There is no consensus on the computability of Graham's number, as some assert it is computable while others disagree.

Contextual Notes

Some claims depend on specific mathematical definitions and assumptions, such as the interpretation of computability and the applicability of Benford's Law. The discussion also highlights unresolved complexities in calculating leading digits for certain large numbers.

BWV
Messages
1,665
Reaction score
1,999
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?
 
  • Like
Likes   Reactions: Delta2
Mathematics news on Phys.org
Why does anybody want to know? This is of similar interest as the 117th digit of ##\pi.##
 
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:
In attempting to do this problem, one might learn new math techniques that could be applied to future problems.
 
  • Like
Likes   Reactions: Delta2
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)
 
  • Like
Likes   Reactions: Delta2
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)##.
 
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​
 
  • Like
Likes   Reactions: Delta2
 
##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).
 
  • Like
  • Informative
Likes   Reactions: dextercioby and BWV
  • #10
Computability has a specific meaning in mathematics: Graham's number is not incomputable.
 
  • Informative
  • Like
Likes   Reactions: mfb and Delta2
  • #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.
 
  • Informative
Likes   Reactions: BWV
  • #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:
 

Similar threads

Undergrad Is dealing with large numbers as formula a thing? and how is it done?
  • · Replies 3 ·
Replies
3
Views
2K
High School Two interesting number theory tricks
  • · Replies 3 ·
Replies
3
Views
1K
MHB Large Prime number unable to compute
  • · Replies 2 ·
Replies
2
Views
8K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad What is the probability that the product of the digits is odd?
  • · Replies 13 ·
Replies
13
Views
3K
Undergrad Calculating the digits of pi by colliding boxes
  • · Replies 3 ·
Replies
3
Views
5K
Undergrad Karatsuba multiplication implementation question
  • · Replies 3 ·
Replies
3
Views
2K
Largest Number Game - Start at 1!
  • · Replies 43 ·
2
Replies
43
Views
6K
Graduate Can a larger base than three create a number greater than Graham's number?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Is this a paradox on the number line
  • · Replies 16 ·
Replies
16
Views
2K