OEIS/3x+1 Problem - Revision history

Gfis: new section

2023-08-02T07:50:56Z

new section

← Older revision		Revision as of 07:50, 2 August 2023
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	~~This article describes an attempt~~ of ~~[mailto:dr.georg.fischer@gmail.com Georg Fischer] to prove~~ the '''[https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture]'''.		A collection of considerations regarding the '''[https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture]'''.

	It is splitted into the following parts:		It is splitted into the following parts:
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	* [[OEIS/3x%2B1_Connectivity]]		* [[OEIS/3x%2B1_Connectivity]]
	* [[OEIS/3x%2B1_Levels]]		* [[OEIS/3x%2B1_Levels]]
			* [[OEIS/Attaching segments]]

imported>Gfis: Levels

2019-08-09T08:09:57Z

Levels

← Older revision		Revision as of 08:09, 9 August 2019
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	This article describes an attempt of [mailto:dr.georg.fischer@gmail.com Georg Fischer] to prove the Collatz conjecture.		This article describes an attempt of [mailto:dr.georg.fischer@gmail.com Georg Fischer] to prove the '''[https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture]'''.

	It is splitted into the following parts:		It is splitted into the following parts:
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	* [[OEIS/3x%2B1_Segments]]		* [[OEIS/3x%2B1_Segments]]
	* [[OEIS/3x%2B1_Connectivity]]		* [[OEIS/3x%2B1_Connectivity]]
	* [[OEIS/3x%~~2B1_Proof~~]]		* [[OEIS/3x%2B1_Levels]]

imported>Gfis: Links to parts only

2019-08-07T18:28:10Z

Links to parts only

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imported>Gfis: before splitting into parts

2019-08-07T18:04:38Z

before splitting into parts

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imported>Gfis at 16:37, 7 August 2019

2019-08-07T16:37:49Z

← Older revision		Revision as of 16:37, 7 August 2019
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	===Segment directory===		===Segment directory===
	The resulting set of small trees is ~~listed~~ in a so-called '''[http://teherba.org/fasces/oeis/collatz/double.html ~~segment directory~~]~~''' (the link shows several thousand subtrees arranged by increasing left side~~, ~~together with some additional information)~~.		The resulting set of small trees is arranged by increasing left sides in a table S with rows of irregular length, the so-called '''segment directory]'''.
	:The segment directory shows the ~~segments in rows~~ of a table. The steps are mapped to columns of the table~~. In order to maintain the subtree structure with two branches, each row contains a pair of columns numbered 3, 5, 7, ... for the upper branch and 2, 4, 6, ... for the lower branch~~.		:The reader is encouraged to examine an online example with the first 800 segments at [http://teherba.org/fasces/oeis/collatz/double.html [http://teherba.org/fasces/oeis/collatz/double.html]. Several variants - which are described later in this article - can be accessed, and trajectories can be followd interactively.
	:Nodes in the branches of a segment which have the form ''6i - 2'' are shown in bold face. If ''i'' also has the form ''6k - 2'', the node has a yellow background.		:The "double line" presentation form of the segment directory shows the two branches of a segment in one table row. The steps are mapped to columns of the table.
			:Nodes in the branches of a segment which have the form ''6i - 2'' are shown in bold face. If ''i'' also has the form ''6k - 2'', the node has a yellow, orange or red background.
	====Details of a segment====		====Details of a segment====
	The following table '''(T1)''' shows various properties of the steps or columns in the segment construction:		The following table '''(T1)''' shows various properties of the steps or columns in the segment construction:
	{\| class="wikitable" style="text-align:left"		{\| class="wikitable" style="text-align:left"
	!Column j \|\| Branch !! Operation !! Form of i !! Formula !! First elements !! Covered ~~nodes~~!! Remaining ~~nodes~~		!Column j \|\| Branch !! Operation !! Form of i !! Formula !! First elements !! Covered residues!! Remaining residues
	\|-		\|-
	\|'''1''' \|\|(LS) \|\| \|\| \|\|6*i - 2 \|\|4, 10, 16, 22 \|\| \|\|1, 2, 4, 5 mod 6		\|'''1''' \|\|(LS) \|\| \|\| \|\|6*i - 2 \|\|4, 10, 16, 22 \|\| \|\|1, 2, 4, 5 mod 6
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	The general formula (in the last row of T1) for a column ''j >= 5'' uses the following parameters:		The general formula (in the last row of T1) for a column ''j >= 5'' uses the following parameters:
	* ''q<sub>j</sub> = 3^(floor((j - 2) / 4)'',		* ''q<sub>j</sub> = 3^(floor((j - 2) / 4)'',
	* ''r<sub>j</sub> = ~~[http://oeis.org/A066443~~ A066443](floor((j - 2) / 8))''~~, where A066443 is~~ the OEIS sequence ~~''a(n)''~~ defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0''; the terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]),		* ''r<sub>j</sub> = A066443(floor((j - 2) / 8))'' with the OEIS sequence [http://oeis.org/A066443 A066443] defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0''; the terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]),
	* ''s<sub>j</sub> = 3*2^floor((j + 7) / 4)'',		* ''s<sub>j</sub> = 3*2^floor((j + 7) / 4)'',
	* ''t<sub>j</sub> = [http://oeis.org/draft/A309523 A309523]~~(j) for j >= 3'', where A309523 is an OEIS sequence computable by a linear recurrence with signature (1, -1, 1, 0, 0, 0, 0, 4, -4, 4, -4)~~.		* ''t<sub>j</sub> = A309523(j) for j >= 3'' with the linear recurrent OEIS sequence [http://oeis.org/draft/A309523 A309523].

	===Finite, arbitrary length of the segments===		===Finite, arbitrary length of the segments===
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	The left sides are in arithmetic progression, and a specific column is computed by a fixed combination of δ and µ operations. Therefore, a column is also in a simple, arithmetic progression. The modulo conditions are all different.		The left sides are in arithmetic progression, and a specific column is computed by a fixed combination of δ and µ operations. Therefore, a column is also in a simple, arithmetic progression. The modulo conditions are all different.
	The initial terms ''t<sub>j</sub>'' from OEIS [https://oeis.org/draft/A309523 A309523] are also disjoint.		The initial terms ''t<sub>j</sub>'' from OEIS [https://oeis.org/draft/A309523 A309523] are also disjoint.
	:The terms of OEIS [https://oeis.org/draft/A308709 A308709] are all disjoint, since that sequence is the set ''{2^k} union {22^k~~} for~~ k > 0'', and OEIS [https://oeis.org/draft/A309523 A309523] is the ~~superset~~ ''{6A308709 - 2} union {6*A308709 - 1}''.		:The terms of OEIS [https://oeis.org/draft/A308709 A308709] are all disjoint, since that sequence is the set ''{2^k \| k>=0} union {32^k \| k>=0}'', and OEIS [https://oeis.org/draft/A309523 A309523] is the set ''{6A308709 - 2} union {6*A308709 - 1}''.
	Together that shows that the terms in the right part are all disjoint.		Together that shows that the terms in the right part are all disjoint.

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	Furthermore, since the modulo conditions lead to disjoint subsets of the numbers, it is ensured that any number can occur only once in the right part.		Furthermore, since the modulo conditions lead to disjoint subsets of the numbers, it is ensured that any number can occur only once in the right part.
	In total, the nodes in the right part are a '''permutation of the numbers''' which are not divisible by 6.		In total, the nodes in the right part are a '''permutation of the numbers''' which are not divisible by 6.
			==Connectivity of the graph==
			With this result we can pick an arbitrary number, start a Collatz trajectory for it, and look in parallel which segments are encountered. We can - easily and uniquely - locate any node of the trajectory in some column in the right part of the segment directory, just be determining the rest modulo 6, 12, 24, 48 ... 62^k and looking it up in T1 resp. OEIS [https://oeis.org/draft/A309523 A309523]. Once the column is known, the segmet directory's row index ''i'' (and thereby the left side ''6i - 2'') can be determined via table T1.

			By locating the trajectory node uniquely in some segment, we "embed" it in a small, local tree structure. The goal is now to show that the segments can be attached ("connected", "glued" ...) to each other such they form a single tree which maps onto the whole Collatz graph.

			For the connectivity considerations we classify all nodes by their residue modulo 6, and we state which of the elementary graph operations δ and µ (away from the root) can be applied to such a node ''i''. We observe the following cases:
			* ''i ≡ 1, 3, 5 mod 6'' (odd node): The next operation must be µ.
			* ''i ≡ 0 mod 6'' (2^k * 3): All following operations must be µ.
			* ''i ≡ 2 mod 6'': The next operation must be µ, and the following node is of the form ''j ≡ 4 mod 6''.
			* ''i ≡ 4 mod 6'': The next operations can be δ or µ.

			==Compressed graph==
			With this permutation of the numbers covering
			When we look back to the Collatz graph we can note that

	==Following a Collatz trajectory==		==Following a Collatz trajectory==
	Since we are sure that all numbers occur once in the right part of our segment directory, we can pick any of them and start a Collatz trajectory, hopefully proceeding down to the root (4). We try to trace the trajectory in the segment directory.		Since we are sure that all numbers occur once in the right part of our segment directory, we can pick any of them and start a Collatz trajectory, hopefully proceeding down to the root (4). We try to trace the trajectory in the segment directory.
	~~----------------------~~
	==Covering the Collatz graph by the right part==		==Covering the Collatz graph by the right part==
	The fact that all numbers occur at exactly one place in the segment directory is a good indication		The fact that all numbers occur at exactly one place in the segment directory is a good indication
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	* that it really has the form of a tree.		* that it really has the form of a tree.

	Any number ''n'', be it a start value or a node in a trajectory, can easily and uniquely be located in some column in the right part just be determining the rest modulo 6, 12, 24, 48 ... 62^k and looking it up in T1 resp. OEIS [https://oeis.org/draft/A309523 A309523]. Once the column is known, the segment directory row index ''i'' (and thereby the left side ''6i - 2'') can also easily be determined via table T1.		Any number ''n'', be it a start value or a

imported>Gfis: trajectory trace

2019-08-06T13:09:11Z

trajectory trace

← Older revision		Revision as of 13:09, 6 August 2019
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	* ''r<sub>j</sub> = [http://oeis.org/A066443 A066443](floor((j - 2) / 8))'', where A066443 is the OEIS sequence ''a(n)'' defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0''; the terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]),		* ''r<sub>j</sub> = [http://oeis.org/A066443 A066443](floor((j - 2) / 8))'', where A066443 is the OEIS sequence ''a(n)'' defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0''; the terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]),
	* ''s<sub>j</sub> = 3*2^floor((j + 7) / 4)'',		* ''s<sub>j</sub> = 3*2^floor((j + 7) / 4)'',
	* ''t<sub>j</sub> = [http://oeis.org/draft/A309523 A309523](j) for j >= 3', where A309523 is an OEIS sequence computable by a linear recurrence with signature (1, -1, 1, 0, 0, 0, 0, 4, -4, 4, -4).		* ''t<sub>j</sub> = [http://oeis.org/draft/A309523 A309523](j) for j >= 3'', where A309523 is an OEIS sequence computable by a linear recurrence with signature (1, -1, 1, 0, 0, 0, 0, 4, -4, 4, -4).

	===Finite, arbitrary length of the segments===		===Finite, arbitrary length of the segments===
	First we have to show that the segment construction process always stops.		First we have to show that the segment construction process always stops.
	For this purpose, we observe that the σ operation, when applied to left sides of the form ''6(3k) - 2'' has the interesting property that it maintains the general form ''6*i - 2'':		For this purpose, we observe that the σ operation, when applied to left sides of the form ''6(3k) - 2'' has the interesting property that it maintains the general form ''6*i - 2'':
	6(3k) - 2 σ (2(3k) - 1) * 2 = 12k - 2 = 6(2*k) - 2		* 6(3k) - 2 σ (2(3k) - 1) * 2 = 12k - 2 = 6(2*k) - 2
	That means that		That means that
	:σ replaces one factor 3 by a factor 2		:'''σ replaces one factor 3 by a factor 2'''
	The mnemonic is that σ "squeezes" a 3 into a 2. In the opposite direction, the ''s ("spike")'' operation replaces one factor 2 by a factor 3.		The mnemonic is that σ "squeezes" a 3 into a 2. In the opposite direction, the ''s ("spike")'' operation replaces one factor 2 by a factor 3.

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	We are now interested how the numbers > 0 are distributed over that right part. Corresponding remarks are contained in the last two columns of table T1. We want to show that the nodes in the right part are a permutation of the numbers > 0.		We are now interested how the numbers > 0 are distributed over that right part. Corresponding remarks are contained in the last two columns of table T1. We want to show that the nodes in the right part are a permutation of the numbers > 0.

	The left sides are in arithmetic progression, and a specific column is computed by a fixed combination of δ and µ operations. Therefore, a column is also in a simple, arithmetic progression. The modulo conditions are all different~~, and so are the~~ initial terms ''t<sub>j</sub>'' from OEIS [https://oeis.org/draft/A309523 A309523]. ~~Togehter~~ that shows that the terms in the right part are all disjoint.		The left sides are in arithmetic progression, and a specific column is computed by a fixed combination of δ and µ operations. Therefore, a column is also in a simple, arithmetic progression. The modulo conditions are all different.
			The initial terms ''t<sub>j</sub>'' from OEIS [https://oeis.org/draft/A309523 A309523] are also disjoint.
			:The terms of OEIS [https://oeis.org/draft/A308709 A308709] are all disjoint, since that sequence is the set ''{2^k} union {22^k} for k > 0'', and OEIS [https://oeis.org/draft/A309523 A309523] is the superset ''{6A308709 - 2} union {6*A308709 - 1}''.
			Together that shows that the terms in the right part are all disjoint.

	So far we did not yet cover numbers of the form ''i ≡ 2, 4 mod 6'' (or ''i ≡ 2, 4, 8, 10 mod 12''). We proceed by looking into each column in the right part and by observing the modulo conditions of the terms contained in that column.		So far we did not yet cover numbers of the form ''i ≡ 2, 4 mod 6'' (or ''i ≡ 2, 4, 8, 10 mod 12''; ''0 mod 6'' was excluded above). We proceed by looking into each column in the right part and by observing the modulo conditions of the terms contained in that column.

	Column 3 contains numbers of the form ''i ≡ 8 mod 12'', ~~so we did not yet cover~~ 2, 4, 10 mod 12. Column 4, if it exists, contains numbers of the form ''i ≡ 2 mod 12''. This leaves us with ''i ≡ 4, 10 mod 12''.		Column 3 contains numbers of the form ''i ≡ 8 mod 12'', such that ''2, 4, 10 mod 12'' remain to be covered. Column 4, if it exists, contains numbers of the form ''i ≡ 2 mod 12''. This leaves us with ''i ≡ 4, 10 mod 12''.

	As indicated in table T1 above, the modulo conditions in the right part become more and more refined. Since the segment directory contains arbitrarily long segments (respectively arbitrarily many columns), the modulo conditions of any height are finally reached.		As indicated in table T1 above, the modulo conditions in the right part become more and more refined. Since the segment directory contains arbitrarily long segments (respectively arbitrarily many columns), the modulo conditions of any height are finally reached.

	Furthermore, since the modulo conditions lead to disjoint subsets of the numbers, it is ensured that any number can occur only once in the right part.		Furthermore, since the modulo conditions lead to disjoint subsets of the numbers, it is ensured that any number can occur only once in the right part.
	In total, the nodes in ~~column 2 together with~~ the ~~terms in the columns which contain even numbers~~ are a '''permutation of the numbers''' which are not divisible by 6.		In total, the nodes in the right part are a '''permutation of the numbers''' which are not divisible by 6.
			==Following a Collatz trajectory==
			Since we are sure that all numbers occur once in the right part of our segment directory, we can pick any of them and start a Collatz trajectory, hopefully proceeding down to the root (4). We try to trace the trajectory in the segment directory.
			----------------------
			==Covering the Collatz graph by the right part==
			The fact that all numbers occur at exactly one place in the segment directory is a good indication
			* that the whole Collatz graph can be covered by the right part of the segment directory,
			* that there are no cycles in the graph, and
			* that it really has the form of a tree.

	~~The fact that all numbers occur at exactly one place~~ in the segment ~~ther is no number which does not occur all nubmers occur in~~ the		Any number ''n'', be it a start value or a node in a trajectory, can easily and uniquely be located in some column in the right part just be determining the rest modulo 6, 12, 24, 48 ... 62^k and looking it up in T1 resp. OEIS [https://oeis.org/draft/A309523 A309523]. Once the column is known, the segment directory row index ''i'' (and thereby the left side ''6i - 2'') can also easily be determined via table T1.

imported>Gfis: with A309523

2019-08-06T11:45:01Z

with A309523

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imported>Gfis: Permutation

2019-08-05T21:34:51Z

Permutation

imported>Gfis: up to coverage

2019-08-05T20:55:34Z

up to coverage

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imported>Gfis: Table T1

2019-08-05T16:35:05Z

Table T1

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← Older revision		Revision as of 21:34, 5 August 2019
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	* to ''(n - 1) / 3'' if ''n ≡ 1 mod 3''.		* to ''(n - 1) / 3'' if ''n ≡ 1 mod 3''.

	The Collatz conjecture claims that the Collatz graph		The '''Collatz conjecture''' claims that the Collatz graph
	* contains all numbers,		* contains all numbers,
	and that it - except for the leading cycle 1 - 2 - 4 - 1 - 2 - 4 ... -		and that - except for the leading cycle 1 - 2 - 4 - 1 - 2 - 4 ... -
	* has the form of a tree (without cycles).		* it has the form of a tree (without cycles).
	We will not consider the leading cycle, and we start the graph with node 4, the '''root'''.		We will not consider the leading cycle, and we start the graph with node 4, the '''root'''.
	Furthermore we observe that a path can only be continued with duplications once it reaches a node ''n ≡ 0 mod 3''. We omit these trivial continuations.		Furthermore we observe that a path can only be continued with duplications once it reaches a node ''n ≡ 0 mod 3''. We omit these trivial continuations.
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	[ 364 µ 728 µ 1456 δ 485 µ 970 δ 323 µ 646 δ 215 µ 430 δ 143 µ 286 δ 95 µ 190 δ 63 ]		[ 364 µ 728 µ 1456 δ 485 µ 970 δ 323 µ 646 δ 215 µ 430 δ 143 µ 286 δ 95 µ 190 δ 63 ]
	[ 364 δ 121 µ 242 µ 484 δ 161 µ 322 δ 107 µ 214 δ 71 µ 142 δ 47 µ 94 δ 31 µ 62 ]		[ 364 δ 121 µ 242 µ 484 δ 161 µ 322 δ 107 µ 214 δ 71 µ 142 δ 47 µ 94 δ 31 µ 62 ]
	===Segments===		==Segments==
	These patterns lead us to the construction of special subtrees in the Collatz graph which we call '''segments'''. Starting at some node of the form ''6*i - 2'' (the ''left side'', LS), two subpaths (the ''upper branch'' and the ''lower branch'') ~~continue~~ to the right by a prescribed sequence of operations.		These patterns lead us to the construction of special subtrees in the Collatz graph which we call '''segments'''. This construction is the main result of this article, and as we will see, it exhibits a nested structure reminding to a ruler. Though the structure is still complicated, it is - in contrast to the Collatz graph - very regular and "predictible" in all it's properties.

			Starting at some node of the form ''6*i - 2'' (the ''left side'', LS), the construction creates two subpaths (the ''upper branch'' and the ''lower branch'') which to the right by a prescribed sequence of operations.
	:The nodes of the form ''6i - 2'' play a special role because both a δ and a µ operation can be applied to them. We could also have used ''6i + 4'' instead, but the formulas seem easier with ''6i - 2'', therefore we always use this form. Later we will consider nodes where ''6i - 2'' is transformed to ''j'', where ''j'' also may have the form ''j = 6*k - 2'', etc.		:The nodes of the form ''6i - 2'' play a special role because both a δ and a µ operation can be applied to them. We could also have used ''6i + 4'' instead, but the formulas seem easier with ''6i - 2'', therefore we always use this form. Later we will consider nodes where ''6i - 2'' is transformed to ''j'', where ''j'' also may have the form ''j = 6*k - 2'', etc.

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	* In a loop, apply opposite operations to both branches, starting with µ (step 6) on the lower and δ (step 7) on the upper branch, continuing with δ (step 8) on the lower and µ (step 9) on the upper branch, and so on, as long as the next δ operation is possible.		* In a loop, apply opposite operations to both branches, starting with µ (step 6) on the lower and δ (step 7) on the upper branch, continuing with δ (step 8) on the lower and µ (step 9) on the upper branch, and so on, as long as the next δ operation is possible.
	The construction is such that increasing instances of σ operations are accumulated in each branch.		The construction is such that increasing instances of σ operations are accumulated in each branch.

	===Segment directory===		===Segment directory===
	The resulting set of small trees is listed in a so-called '''[http://teherba.org/fasces/oeis/collatz/double.html segment directory]''' (the link shows several thousand subtrees arranged by increasing left side, together with some additional information).		The resulting set of small trees is listed in a so-called '''[http://teherba.org/fasces/oeis/collatz/double.html segment directory]''' (the link shows several thousand subtrees arranged by increasing left side, together with some additional information).
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	\|11 \|\|upper\|\|µ<sup>2</sup>σδ \|\|i = 9k + 7 \|\|48(i - 7)/9 + 35 \|\|35, 83, 131, 179 \|\|(cf. column 2)\|\|		\|11 \|\|upper\|\|µ<sup>2</sup>σδ \|\|i = 9k + 7 \|\|48(i - 7)/9 + 35 \|\|35, 83, 131, 179 \|\|(cf. column 2)\|\|
	\|-		\|-
	\|12 \|\|lower\|\|σµσδ \|\|i = 9k + 7 \|\|48(i - 7)/9 + 11 \|\|11, 27, 43, 59 \|\|(cf. column 2)\|\|		\|12 \|\|lower\|\|σµσδ \|\|i = 9k + 7 \|\|48(i - 7)/9 + 11 \|\|11, 59, 107, 155 \|\|(cf. column 2)\|\|
	\|-		\|-
	\|'''13'''\|\|upper\|\|µ<sup>2</sup>σ<sup>2</sup> \|\|i = 9k + 7 \|\|96(i - 7)/9 + 70 \|\|70, 166, 262, 358 \|\|70 mod 96 \|\|22, 46, 94 mod 96		\|'''13'''\|\|upper\|\|µ<sup>2</sup>σ<sup>2</sup> \|\|i = 9k + 7 \|\|96(i - 7)/9 + 70 \|\|70, 166, 262, 358 \|\|70 mod 96 \|\|22, 46, 94 mod 96
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	\|16 \|\|lower\|\|σµσ<sup>2</sup>δ \|\|i = 27k + 7 \|\|96(i - 7)/27 + 71 \|\|71, 167, 263, 359 \|\|(cf. column 2) \|\|		\|16 \|\|lower\|\|σµσ<sup>2</sup>δ \|\|i = 27k + 7 \|\|96(i - 7)/27 + 71 \|\|71, 167, 263, 359 \|\|(cf. column 2) \|\|
	\|-		\|-
	\|'''17'''\|\|upper\|\|µ<sup>2</sup>σ<sup>3</sup> \|\|i = 27k + 7 \|\|192(i - 7)/27 + 46 \|\|46, 238, 430, 622 \|\|46 mod 96 \|\|94, 142, 190 mod 192		\|'''17'''\|\|upper\|\|µ<sup>2</sup>σ<sup>3</sup> \|\|i = 27k + 7 \|\|192(i - 7)/27 + 46 \|\|46, 238, 430, 622 \|\|46 mod 192 \|\|94, 142, 190 mod 192
	\|-		\|-
	\|'''18'''\|\|lower\|\|σµσ<sup>3</sup> \|\|i = 81k + 61\|\|192(i - 61)/81 + 142 \|\|142, 334, 526, 718\|\|142 mod 96 \|\|94, 190, 286, 382 mod 384		\|'''18'''\|\|lower\|\|σµσ<sup>3</sup> \|\|i = 81k + 61\|\|192(i - 61)/81 + 142 \|\|142, 334, 526, 718\|\|142 mod 192 \|\|94, 190, 286, 382 mod 384
	\|-		\|-
	\|19 \|\|upper\|\|µ<sup>2</sup>σ<sup>3</sup>δ \|\|i = 81k + 61\|\|192(i - 61)/81 + 143 \|\|143, 225, 527, 719\|\|(cf. column 2) \|\|		\|19 \|\|upper\|\|µ<sup>2</sup>σ<sup>3</sup>δ \|\|i = 81k + 61\|\|192(i - 61)/81 + 143 \|\|143, 225, 527, 719\|\|(cf. column 2) \|\|
	\|-		\|-
	\| j \|\| \|\| \|\|i = q<sub>j</sub>k + r<sub>j</sub> \|\| s<sub>j</sub>(i - r<sub>j</sub>)/q<sub>j</sub> + t<sub>j</sub> \|\| u<sub>j</sub>, ... \|\| \|\|		\| j \|\| \|\| \|\|i = q<sub>j</sub>k + r<sub>j</sub> \|\| s<sub>j</sub>(i - r<sub>j</sub>)/q<sub>j</sub> + t<sub>j</sub> \|\| t<sub>j</sub>, ... \|\| t<sub>j</sub> mod s<sub>j</sub> \|\|
	\|-		\|-
	\|}		\|}
	The general formula (in the last row of T1) for a column ''j >= 5'' uses the following parameters:		The general formula (in the last row of T1) for a column ''j >= 5'' uses the following parameters:
	* ''q<sub>j</sub> = 3^(floor((j - 2) / 4)''		* ''q<sub>j</sub> = 3^(floor((j - 2) / 4)'',
	* ''r<sub>j</sub> = [http://oeis.org/A066443 A066443(floor((j - 2) / 8)]'', where A066443 is the OEIS sequence ''a(n)'' defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0'' ~~. The~~ terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]).		* ''r<sub>j</sub> = [http://oeis.org/A066443 A066443](floor((j - 2) / 8))'', where A066443 is the OEIS sequence ''a(n)'' defined by ''a(0) = 1; a(n) = 9 * a(n-1) - 2 for n > 0''; the terms are the indexes 1, 7, 61, 547, 4921 ... of the variable length segments with left sides 4, 40, 364, 3280, 29524 ([http://oeis.org/A191681 OEIS A191681]),
	* ''s<sub>j</sub = 3*2^floor((j + 7) / 4)''		* ''s<sub>j</sub> = 3*2^floor((j + 7) / 4)'',
	* ''t<sub>j</sub = 6*([http://oeis.org/draft/A308709 A308709(j - 4)] - 2'', where A308709 = 3, 1, 2, 6, 12, 4, ... a simple, linear recurrent OEIS sequence		* ''t<sub>j</sub> = 6[http://oeis.org/draft/A308709 A308709](floor(j + 2floor((j - 3) / 4))/3)) - 2'', where A308709 = 3, 1, 2, 6, 12, 4, ... is a simple, linear recurrent OEIS sequence (only for j = 5,6, 9,10, 13,14, ...).
	===Finite, arbitrary length of the segments===		===Finite, arbitrary length of the segments===
	First we have to show that the segment construction process always stops.		First we have to show that the segment construction process always stops.