The Syracuse conjecture for U(0)={1,2,3,4,...,256} -monodimensional display- [La conjecture de Syracuse pour U(0)={1,2,3,4,...,256} -visualisation monodimensionnelle-].

• The Syracuse sequence is defined as follows:

## U = N (an integer number [un nombre entier]) > 0 0 if U is even [si U est pair] : n n U n U = ---- n+1 2 else [sinon] : U = 3*U + 1 n+1 n

• The Syracuse conjecture states that sooner or later the {[[4,] 2,] 1} sequence will appear whatever the starting number N (and then repeats itself obviously ad vitam aeternam). For example with U(0)=7:
U(0)  =  7
U(1)  = 22
U(2)  = 11
U(3)  = 34
U(4)  = 17
U(5)  = 52
U(6)  = 26
U(7)  = 13
U(8)  = 40
U(9)  = 20
U(10) = 10
U(11) =  5
U(12) = 16
U(13) =  8
U(14) =  4
U(15) =  2
U(16) =  1

• 256 Syracuse sequences are computed from U(0)=1 to U(0)=256 (=10 in the following explanations for the sake of simplicity). Each horizontal line displays the sequences {U(n)} (each in reverse order) for U(0)=1 to U(0)=10, U(0) being the last number on each line (each U(0) is followed with the percentage of even U(n)):

U(0) =   1  (0.00%) : {U(n)}={0001}
U(0) =   2 (50.00%) : {U(n)}={0001,0002}
U(0) =   3 (62.50%) : {U(n)}={0001,0002,0004,0008,0016,0005,0010,0003}
U(0) =   4 (66.67%) : {U(n)}={0001,0002,0004}
U(0) =   5 (66.67%) : {U(n)}={0001,0002,0004,0008,0016,0005}
U(0) =   6 (66.67%) : {U(n)}={0001,0002,0004,0008,0016,0005,0010,0003,0006}
U(0) =   7 (64.71%) : {U(n)}={0001,0002,0004,0008,0016,0005,0010,0020,0040,0013,0026,0052,0017,0034,0011,0022,0007}
U(0) =   8 (75.00%) : {U(n)}={0001,0002,0004,0008}
U(0) =   9 (65.00%) : {U(n)}={0001,0002,0004,0008,0016,0005,0010,0020,0040,0013,0026,0052,0017,0034,0011,0022,0007,0014,0028,0009}
U(0) =  10 (71.43%) : {U(n)}={0001,0002,0004,0008,0016,0005,0010}

Then the sequences are sorted in order to exhibit a tree:

0001 0002 0004 0008 0016 0005 0010 0020 0040 0013 0026 0052 0017 0034 0011 0022 0007 0014 0028 0009
0001 0002 0004 0008 0016 0005 0010 0020 0040 0013 0026 0052 0017 0034 0011 0022 0007
0001 0002 0004 0008 0016 0005 0010 0003 0006
0001 0002 0004 0008 0016 0005 0010 0003
0001 0002 0004 0008 0016 0005 0010
0001 0002 0004 0008 0016 0005
0001 0002 0004 0008
0001 0002 0004
0001 0002
0001

The first node -root-, the second one and the third one are respectively "0001", "0002" and "0004" (these three nodes being obviously common to all the sequences without exception if the conjecture is true!):

--> 0014 0028 0009
|
--> 0020 0040 0013 0026 0052 0017 0034 0011 0022 0007
|
|     --> 0006
|    |
|--> 0003
|
--> 0010
|
--> 0016 0005
|
--> 0008
|
--> 0004
|
--> 0002
|
0001

The luminance of each node is an increasing function of U(n). See the colors used (the smallest U(n) -equals to 1- is on the left hand side when the biggest -equals to 13120- is on the right hand side):

One can notice the high percentage of even U(n). It is obvious for on the one hand 3*U(n)+1 is even when U(n) is odd. On the other hand, if U(n) is even then U(n)/2 is even with a 0.5 probability. Then there should be more even numbers that odd ones...

See some related visualizations (including this one):

(CMAP28 WWW site: this page was created on 09/18/2017 and last updated on 01/30/2023 11:32:14 -CET-)

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