The Syracuse conjecture for U(0)={5,6,7,8,...,68} -polar coordinates display- [La conjecture de Syracuse pour U(0)={5,6,7,8,...,68} -visualisation en coordonnées polaires-].
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
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
Rho(n) = U (with a renormalization inside [0,1]) n
2.pi Theta(n) = ------.n nM+1
X(n) = Rho(n).cos(Theta(n)) Y(n) = Rho(n).sin(Theta(n))where 'nM' denotes the first 'n' such as:
U = 1 nMThe colors used are a function of 'n' (from Dark Blue [n=0] to White with an increasing luminance ).