AVirtualMachineForExploringSpaceTimeAndBeyond : The Syracuse conjecture for U(0)={5,10,15,20,...,640} -monodimensional display- (La conjecture de Syracuse pour U(0)={5,10,15,20,...,640} -visualisation monodimensionnelle-).

The Syracuse conjecture for U(0)={5,10,15,20,...,640} -monodimensional display- [La conjecture de Syracuse pour U(0)={5,10,15,20,...,640} -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:

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

The horizontal and vertical axes represent the integer numbers {5, 10, 15, 20,...}. 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'.

See some related visualizations (including this one):