The Syracuse conjecture for U(0)={1,2,3,4,...,128} -monodimensional display of the parities- [*La conjecture de Syracuse pour U(0)={1,2,3,4,...,128} -visualisation monodimensionnelle des parités-*].

- 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 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.

See some related visualizations (including this one):

(CMAP28 WWW site: this page was created on 04/18/2019 and last updated on 02/28/2022 11:11:48 -CET-)

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