AVirtualMachineForExploringSpaceTimeAndBeyond : The Syracuse conjecture for U(0)={1,2,3,4,...,262144} (La conjecture de Syracuse pour U(0)={1,2,3,4,...,262144}).

The Syracuse conjecture for U(0)={1,2,3,4,...,262144} [La conjecture de Syracuse pour U(0)={1,2,3,4,...,262144}].

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.

Starting from the origin of the coordinates (at the center of the picture), one follows a square spiral-like path and numbers each integer point encountered (1, 2, 3,...):

                    5----4----3
                    |         |    .
                    |         |    .
                    6    1----2    .
                    |              |
                    |              |
                    7----8----9----10

At last, display the N-th point with the false color f(tst(N)) where:

                    tst(N) = number of iterations of the Syracuse sequence starting with N before reaching the first {[[4,] 2,] 1} sequence (or Total Stopping Time).

See some related visualizations (including this one):