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**

Here are 256 different sequences starting from U(0)=1 to U(0)=256.

- The horizontal and vertical axes represent the integer numbers {1, 2, 3, 4,...}.
Each vertical line (with abscissa equals to N) displays the sequence U(n) starting at U(0)=N
and the color of each of its points {N,U(n)} is a function of 'N'. For example with N=7,
the X=7 vertical line is displayed by means of the 17 following points:
{7,7} C=7 {7,22} C=7 {7,11} C=7 {7,34} C=7 {7,17} C=7 {7,52} C=7 {7,26} C=7 {7,13} C=7 {7,40} C=7 {7,20} C=7 {7,10} C=7 {7,5} C=7 {7,16} C=7 {7,8} C=7 {7,4} C=7 {7,2} C=7 {7,1} C=7

where 'C' denotes the color of the points.

See some related visualizations (including this one):

(CMAP28 WWW site: this page was created on 01/15/2013 and last updated on 02/28/2022 11:11:29 -CET-)

[See all related pictures (including this one) [

[Please visit the related NumberTheory picture gallery [

[

[Mail [

[About Pictures and Animations [

Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2022.