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A006577: Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.
If n is 0 or negative, then A006577(n) = -1.
0: 0, 0, 0, 0, 0, 0, 0, ...
-1: -2, -1, -2, -1, -2, -1, ...
-3: -8, -4, -2, -1, -2, -1, -2, -1, ...
-5: -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, ...
-6: -3, -8, -4, -2, -1, -2, -1, -2, -1, ...
-9: -26, -13, -38, -19, -56, -28, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, ... 2A00:6020:A123:8B00:3913:1297:6B6B:CCEF ( talk) 13:19, 14 December 2023 (UTC)
This article could have used an animation for trinary numbers, say, a .GIF of 4975 being broken down. 81.89.66.133 ( talk) 13:01, 2 February 2024 (UTC)
This function is mentioned in the French page and avoid testing parity. [ [1]] Japarthur ( talk) 09:02, 8 March 2024 (UTC)
This
edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
BEFORE: Eliahou (1993) proved that the period p of any non-trivial cycle is of the form AFTER: Eliahou (1993) proved that the period p of the next candidate for a non-trivial cycle is of the form
Idk, im not a mathematician, but i read the paper cited and the p that is used here seems to be only the current (1993) best candidate for a loop above m=2**39, but below 2**48 . As it is earlier mentioned, this region has already been investigated, so not only the original statements "any" false, but the big letter equation is also irrelevant. 89.223.151.22 ( talk) 07:55, 16 March 2024 (UTC)
Collatz Conjecture Solution The Collatz conjecturela is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. /info/en/?search=Collatz conjecture The solution in simple words, all number made out of 1. Like 1=1 2=1+1 3=1+1+1 4=1+1+1+1| 5=1+1+1+1+1 Etc. I am saying not only 1 is repetitive but 4,2,1 is repetitive. 3x+1 in if x=1 then, 3(1)+1= 4, then as per rules 4/2 =2 then 2/2=1 means 4,2,1 Now if x=2 then 3(2)+1=7 then as per rules 3(7)+1=22, then 22/2= 11, then 3(11)+1= 34 then 34/2= 16, 16/2=8, 8/2=4, 4/2=2, 2/2=1. Now if x=3 then 3(3)+1=10, 10/2 = 5, 3(5)+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1| If we see in all solutions starting from one of the small integers 4,2,1 is repetitive. Because in x=3 there is ans 5. allAll ISSN:3006-4023 (Online),JournalofArtificiallntelligence GeneralScience (JAIGS)86 GaurangkumarPatel20894 ( talk) 07:01, 5 April 2024 (UTC)
This
edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
Explaining Convergence
While a mathematical formula or explanation-based proof may yet be difficult, it is possible to explain the underlying mechanism responsible for convergence by transforming the problem statement as follows Collatz Conjecture - Explaining the Convergence: a) For an odd number N, (3N+1) is always an even number, therefore the next step will always be a division by 2. Both these steps can be considered as a single operation, i.e. (3N+1)/2.
b) Sequential multiplication steps (3N+1)/2 may be considered as a single operation until an even number is obtained.
c) Sequential division (N/2) may be considered as a single operation until an odd number is obtained.
With the help of these transformations, it can be observed that for a starting odd number series, (2i-1)*2n-1,
a) Multiplication steps result in the even series (2i-1)*3n-1
b) The resulting superset of even values is represented by either {(6i-4), i ∈ ℕ} or by {(6i-1)*3n-1 and (6i-5)*3n-1, i & n ∈ ℕ}
c) The next set of odd numbers obtained through division steps is represented by {(6i-5)*22n-1]/3 and (6i-1)*22n+1-1]/3, i & n ∈ ℕ}
While mathematical traceability between the starting odd number and the next odd number obtained in step (c) above is difficult to maintain, this approach still helps understand the underlying mechanism leading to convergence.
To explain this, one may begin from the other end of the problem and perform (2N-1)/3 operations on an even series (instead of (3N+1)/2 on a starting odd series). The starting even series E1=1*22n-1, results in a set of odd numbers which can be multiplied by 2n or 2n-1 if the odd number belongs to series {6i-5} or {6i-1} respectively. Sequential performance of these inverse operations results in a hierarchy of even series as shown in this exhibit Hierarchy of Even Series. Rakesh Vajpai ( talk) 06:22, 8 April 2024 (UTC)
The project math101.guru/en/category/collatz/ contains the logs of the Collatz function for some of the top known megaprimes and their vicinities (can not add link due to spam filter). The data currently cover 15 out of Top 17 known megaprimes (except for #7 and #8 discovered in 2023). There are also some interesting graphs with numerical data available that may be added to other graphs in the article. As far as I could say, such big numbers (>45 in total) have never been tested for the validity of the Collatz conjecture. Re2000 ( talk) 07:06, 14 April 2024 (UTC)
This is the
talk page for discussing improvements to the
Collatz conjecture article. This is not a forum for general discussion of the article's subject. |
Article policies
|
Find sources: Google ( books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL |
Archives: 1, 2, 3Auto-archiving period: 50 days |
This
level-5 vital article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
Daily pageviews of this article
A graph should have been displayed here but
graphs are temporarily disabled. Until they are enabled again, visit the interactive graph at
pageviews.wmcloud.org |
A006577: Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.
If n is 0 or negative, then A006577(n) = -1.
0: 0, 0, 0, 0, 0, 0, 0, ...
-1: -2, -1, -2, -1, -2, -1, ...
-3: -8, -4, -2, -1, -2, -1, -2, -1, ...
-5: -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, ...
-6: -3, -8, -4, -2, -1, -2, -1, -2, -1, ...
-9: -26, -13, -38, -19, -56, -28, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, ... 2A00:6020:A123:8B00:3913:1297:6B6B:CCEF ( talk) 13:19, 14 December 2023 (UTC)
This article could have used an animation for trinary numbers, say, a .GIF of 4975 being broken down. 81.89.66.133 ( talk) 13:01, 2 February 2024 (UTC)
This function is mentioned in the French page and avoid testing parity. [ [1]] Japarthur ( talk) 09:02, 8 March 2024 (UTC)
This
edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
BEFORE: Eliahou (1993) proved that the period p of any non-trivial cycle is of the form AFTER: Eliahou (1993) proved that the period p of the next candidate for a non-trivial cycle is of the form
Idk, im not a mathematician, but i read the paper cited and the p that is used here seems to be only the current (1993) best candidate for a loop above m=2**39, but below 2**48 . As it is earlier mentioned, this region has already been investigated, so not only the original statements "any" false, but the big letter equation is also irrelevant. 89.223.151.22 ( talk) 07:55, 16 March 2024 (UTC)
Collatz Conjecture Solution The Collatz conjecturela is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. /info/en/?search=Collatz conjecture The solution in simple words, all number made out of 1. Like 1=1 2=1+1 3=1+1+1 4=1+1+1+1| 5=1+1+1+1+1 Etc. I am saying not only 1 is repetitive but 4,2,1 is repetitive. 3x+1 in if x=1 then, 3(1)+1= 4, then as per rules 4/2 =2 then 2/2=1 means 4,2,1 Now if x=2 then 3(2)+1=7 then as per rules 3(7)+1=22, then 22/2= 11, then 3(11)+1= 34 then 34/2= 16, 16/2=8, 8/2=4, 4/2=2, 2/2=1. Now if x=3 then 3(3)+1=10, 10/2 = 5, 3(5)+1=16, 16/2=8, 8/2=4, 4/2=2, 2/2=1| If we see in all solutions starting from one of the small integers 4,2,1 is repetitive. Because in x=3 there is ans 5. allAll ISSN:3006-4023 (Online),JournalofArtificiallntelligence GeneralScience (JAIGS)86 GaurangkumarPatel20894 ( talk) 07:01, 5 April 2024 (UTC)
This
edit request has been answered. Set the |answered= or |ans= parameter to no to reactivate your request. |
Explaining Convergence
While a mathematical formula or explanation-based proof may yet be difficult, it is possible to explain the underlying mechanism responsible for convergence by transforming the problem statement as follows Collatz Conjecture - Explaining the Convergence: a) For an odd number N, (3N+1) is always an even number, therefore the next step will always be a division by 2. Both these steps can be considered as a single operation, i.e. (3N+1)/2.
b) Sequential multiplication steps (3N+1)/2 may be considered as a single operation until an even number is obtained.
c) Sequential division (N/2) may be considered as a single operation until an odd number is obtained.
With the help of these transformations, it can be observed that for a starting odd number series, (2i-1)*2n-1,
a) Multiplication steps result in the even series (2i-1)*3n-1
b) The resulting superset of even values is represented by either {(6i-4), i ∈ ℕ} or by {(6i-1)*3n-1 and (6i-5)*3n-1, i & n ∈ ℕ}
c) The next set of odd numbers obtained through division steps is represented by {(6i-5)*22n-1]/3 and (6i-1)*22n+1-1]/3, i & n ∈ ℕ}
While mathematical traceability between the starting odd number and the next odd number obtained in step (c) above is difficult to maintain, this approach still helps understand the underlying mechanism leading to convergence.
To explain this, one may begin from the other end of the problem and perform (2N-1)/3 operations on an even series (instead of (3N+1)/2 on a starting odd series). The starting even series E1=1*22n-1, results in a set of odd numbers which can be multiplied by 2n or 2n-1 if the odd number belongs to series {6i-5} or {6i-1} respectively. Sequential performance of these inverse operations results in a hierarchy of even series as shown in this exhibit Hierarchy of Even Series. Rakesh Vajpai ( talk) 06:22, 8 April 2024 (UTC)
The project math101.guru/en/category/collatz/ contains the logs of the Collatz function for some of the top known megaprimes and their vicinities (can not add link due to spam filter). The data currently cover 15 out of Top 17 known megaprimes (except for #7 and #8 discovered in 2023). There are also some interesting graphs with numerical data available that may be added to other graphs in the article. As far as I could say, such big numbers (>45 in total) have never been tested for the validity of the Collatz conjecture. Re2000 ( talk) 07:06, 14 April 2024 (UTC)