AVirtualMachineForExploringSpaceTimeAndBeyond : The Syracuse conjecture for U(0)={5,6,7,8,...,260} -'pseudo-ramdom walk' display : 'The Syracuse Supernova'- (La conjecture de Syracuse pour U(0)={5,6,7,8,...,260} -visualisation sous forme d'une 'pseudo-marche aleatoire' : 'La Supernova de Syracuse'-).

The Syracuse conjecture for U(0)={5,6,7,8,...,260} -'pseudo-ramdom walk' display: 'The Syracuse Supernova'- [La conjecture de Syracuse pour U(0)={5,6,7,8,...,260} -visualisation sous forme d'une 'pseudo-marche aléatoire': 'La Supernova de Syracuse'-].

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:

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

This picture is a circular display of 256 different sequences from U(0)=5 (on the right-hand side) to U(0)=260 -rotating counterclockwise-. For each sequence {U(n)} a "pseudo-ramdom walk" is generated: the parities {0,1} of each U(n) define two angles {-THETA,+THETA} at the center. The 256 starting U(0) values are located at the periphery of the structure -dark orange, with a radius proportional to the length of the sequence {U(n)}-, when the 256 {4,2,1} sequences are at the center -white-.

See some related visualizations (including this one):