The Syracuse Conjecture

(the parities of the Syracuse sequence for the numbers 1 to 128)

Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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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 {1, 2, 3, 4,...}. Each vertical line (with abscissa equals to N) displays the parity of each U(n) starting at U(0)=N and the color of each of its points {N,U(n)} is function of the parity of 'U(n)' (Red=even, White=odd). For example with N=7, the X=7 vertical line is displayed by means of the 17 following points:

                    {7,7}  P=1 (White)
                    {7,22} P=0 (Red)
                    {7,11} P=1 (White)
                    {7,34} P=0 (Red)
                    {7,17} P=1 (White)
                    {7,52} P=0 (Red)
                    {7,26} P=0 (Red)
                    {7,13} P=1 (White)
                    {7,40} P=0 (Red)
                    {7,20} P=0 (Red)
                    {7,10} P=0 (Red)
                    {7,5}  P=1 (White)
                    {7,16} P=0 (Red)
                    {7,8}  P=0 (Red)
                    {7,4}  P=0 (Red)
                    {7,2}  P=0 (Red)
                    {7,1}  P=1 (White)

where 'P' denotes the parity.

One can concatenate all the parities {P} giving the following binary number:

                    10000100010010101

that is 67733 as a decimal number.

Here are the parities -as decimal and then as binary numbers- of 128 different sequences starting from U(0)=1 to U(0)=128.

                    U(0) = 001          -->                                         1     1
                    U(0) = 002          -->                                         2     10
                    U(0) = 003          -->                                       133     10000101
                    U(0) = 004          -->                                         4     100
                    U(0) = 005          -->                                        33     100001
                    U(0) = 006          -->                                       266     100001010
                    U(0) = 007          -->                                     67733     10000100010010101
                    U(0) = 008          -->                                         8     1000
                    U(0) = 009          -->                                    541865     10000100010010101001
                    U(0) = 010          -->                                        66     1000010
                    U(0) = 011          -->                                     16933     100001000100101
                    U(0) = 012          -->                                       532     1000010100
                    U(0) = 013          -->                                       529     1000010001
                    U(0) = 014          -->                                    135466     100001000100101010
                    U(0) = 015          -->                                    135253     100001000001010101
                    U(0) = 016          -->                                        16     10000
                    U(0) = 017          -->                                      4233     1000010001001
                    U(0) = 018          -->                                   1083730     100001000100101010010
                    U(0) = 019          -->                                   1083717     100001000100101000101
                    U(0) = 020          -->                                       132     10000100
                    U(0) = 021          -->                                       129     10000001
                    U(0) = 022          -->                                     33866     1000010001001010
                    U(0) = 023          -->                                     33813     1000010000010101
                    U(0) = 024          -->                                      1064     10000101000
                    U(0) = 025          -->                                   8669737     100001000100101000101001
                    U(0) = 026          -->                                      1058     10000100010
                    U(0) = 027          -->        2678946987458595510314019806849701     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010010010101010100101
                    U(0) = 028          -->                                    270932     1000010001001010100
                    U(0) = 029          -->                                    270929     1000010001001010001
                    U(0) = 030          -->                                    270506     1000010000010101010
                    U(0) = 031          -->          83717093358081109697313118964053     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010101
                    U(0) = 032          -->                                        32     100000
                    U(0) = 033          -->                                  69357897     100001000100101000101001001
                    U(0) = 034          -->                                      8466     10000100010010
                    U(0) = 035          -->                                      8453     10000100000101
                    U(0) = 036          -->                                   2167460     1000010001001010100100
                    U(0) = 037          -->                                   2167457     1000010001001010100001
                    U(0) = 038          -->                                   2167434     1000010001001010001010
                    U(0) = 039          -->                               17755619989     10000100010010100010100001010010101
                    U(0) = 040          -->                                       264     100001000
                    U(0) = 041          -->         669736746864648877578504951712425     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010101001
                    U(0) = 042          -->                                       258     100000010
                    U(0) = 043          -->                                 554869029     100001000100101010000100100101
                    U(0) = 044          -->                                     67732     10000100010010100
                    U(0) = 045          -->                                     67729     10000100010010001
                    U(0) = 046          -->                                     67626     10000100000101010
                    U(0) = 047          -->          20929273339520277424328279741013     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101
                    U(0) = 048          -->                                      2128     100001010000
                    U(0) = 049          -->                                  17339657     1000010001001010100001001
                    U(0) = 050          -->                                  17339474     1000010001001010001010010
                    U(0) = 051          -->                                  17339461     1000010001001010001000101
                    U(0) = 052          -->                                      2116     100001000100
                    U(0) = 053          -->                                      2113     100001000001
                    U(0) = 054          -->        5357893974917191020628039613699402     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010101001010
                    U(0) = 055          -->        5357893974917191020628039613699349     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010100010101
                    U(0) = 056          -->                                    541864     10000100010010101000
                    U(0) = 057          -->                                4438952233     100001000100101010000100100101001
                    U(0) = 058          -->                                    541858     10000100010010100010
                    U(0) = 059          -->                                4438904997     100001000100101000101000010100101
                    U(0) = 060          -->                                    541012     10000100000101010100
                    U(0) = 061          -->                                    541009     10000100000101010001
                    U(0) = 062          -->         167434186716162219394626237928106     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101010
                    U(0) = 063          -->         167434186716162219394626237924693     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101000010101010101
                    U(0) = 064          -->                                        64     1000000
                    U(0) = 065          -->                                 138717257     1000010001001010100001001001
                    U(0) = 066          -->                                 138715794     1000010001001010001010010010
                    U(0) = 067          -->                                 138715781     1000010001001010001010000101
                    U(0) = 068          -->                                     16932     100001000100100
                    U(0) = 069          -->                                     16929     100001000100001
                    U(0) = 070          -->                                     16906     100001000001010
                    U(0) = 071          -->           5232318334880069356082069935253     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010010010101
                    U(0) = 072          -->                                   4334920     10000100010010101001000
                    U(0) = 073          -->       42863151799337528165024316909594793     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010100010101001
                    U(0) = 074          -->                                   4334914     10000100010010101000010
                    U(0) = 075          -->                                     16421     100000000100101
                    U(0) = 076          -->                                   4334868     10000100010010100010100
                    U(0) = 077          -->                                   4334865     10000100010010100010001
                    U(0) = 078          -->                               35511239978     100001000100101000101000010100101010
                    U(0) = 079          -->                               35511239765     100001000100101000101000010001010101
                    U(0) = 080          -->                                       528     1000010000
                    U(0) = 081          -->                                   4328073     10000100000101010001001
                    U(0) = 082          -->        1339473493729297755157009903424850     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101010010
                    U(0) = 083          -->        1339473493729297755157009903424837     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101000101
                    U(0) = 084          -->                                       516     1000000100
                    U(0) = 085          -->                                       513     1000000001
                    U(0) = 086          -->                                1109738058     1000010001001010100001001001010
                    U(0) = 087          -->                                1109738005     1000010001001010100001000010101
                    U(0) = 088          -->                                    135464     100001000100101000
                    U(0) = 089          -->                                1109726249     1000010001001010001010000101001
                    U(0) = 090          -->                                    135458     100001000100100010
                    U(0) = 091          -->              5109685873906317730548896421     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101
                    U(0) = 092          -->                                    135252     100001000001010100
                    U(0) = 093          -->                                    135249     100001000001010001
                    U(0) = 094          -->          41858546679040554848656559482026     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010010010101010
                    U(0) = 095          -->          41858546679040554848656559481173     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010000101010101
                    U(0) = 096          -->                                      4256     1000010100000
                    U(0) = 097          -->      342905214394700225320194535276758345     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101010100010101001001
                    U(0) = 098          -->                                  34679314     10000100010010101000010010
                    U(0) = 099          -->                                  34679301     10000100010010101000000101
                    U(0) = 100          -->                                  34678948     10000100010010100010100100
                    U(0) = 101          -->                                  34678945     10000100010010100010100001
                    U(0) = 102          -->                                  34678922     10000100010010100010001010
                    U(0) = 103          -->               159677683559572429079653013     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101
                    U(0) = 104          -->                                      4232     1000010001000
                    U(0) = 105          -->                              284089918121     100001000100101000101000010001010101001
                    U(0) = 106          -->                                      4226     1000010000010
                    U(0) = 107          -->           1308079583720017339020517483813     10000100000101010001000010010010000101010100010101010101001010010101000100101010100101010010100100101
                    U(0) = 108          -->       10715787949834382041256079227398804     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101010010100
                    U(0) = 109          -->       10715787949834382041256079227398801     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101010010001
                    U(0) = 110          -->       10715787949834382041256079227398698     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001001010101000101010
                    U(0) = 111          -->                     609122022856034962005     1000010000010101000100001001001000010101010001010101010100101001010101
                    U(0) = 112          -->                                   1083728     100001000100101010000
                    U(0) = 113          -->                                      4105     1000000001001
                    U(0) = 114          -->                                8877904466     1000010001001010100001001001010010
                    U(0) = 115          -->                                8877904453     1000010001001010100001001001000101
                    U(0) = 116          -->                                   1083716     100001000100101000100
                    U(0) = 117          -->                                   1083713     100001000100101000001
                    U(0) = 118          -->                                8877809994     1000010001001010001010000101001010
                    U(0) = 119          -->                                8877809941     1000010001001010001010000100010101
                    U(0) = 120          -->                                   1082024     100001000001010101000
                    U(0) = 121          -->             40877486991250541844391171369     100001000001010100010000100100100001010101000101010101010010100101010001001010101001010100101001
                    U(0) = 122          -->                                   1082018     100001000001010100010
                    U(0) = 123          -->                            72727019472037     10000100010010100010100001010010100010010100101
                    U(0) = 124          -->         334868373432324438789252475856212     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010010010101010100
                    U(0) = 125          -->         334868373432324438789252475856209     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010010010101010001
                    U(0) = 126          -->         334868373432324438789252475849386     1000010000010101000100001001001000010101010001010101010100101001010100010010101010010101001010000101010101010
                    U(0) = 127          -->                            72726967522645     10000100010010100010001000100001001010101010101
                    U(0) = 128          -->                                       128     10000000

By the way one cannot find the following binary sequence "11". As a matter of fact, if U(n) is odd (P=1) it means that:

                    U  = 2.k + 1
                     n

and:

                    U    = 3.U  + 1 = 3.(2.k + 1) + 1 = 6.k + 4 = 2.(3.k + 2)
                     n+1      n

Then U(n+1) is even (P=0):
U odd ==> U even n n+1

The Syracuse Conjecture (the parities of the Syracuse sequence for the numbers 1 to 128)

Jean-François COLONNA [Contact me]

www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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

Copyright © Jean-François COLONNA, 2019-2023. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2019-2023.

The Syracuse Conjecture

(the parities of the Syracuse sequence for the numbers 1 to 128)

Jean-François COLONNA
[Contact me]

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

Copyright © Jean-François COLONNA, 2019-2023.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2019-2023.