A276125 a(n) = maximum eventual period of z := z^2 + c (mod n), for Gaussian integers z, c.
1, 2, 3, 2, 6, 6, 12, 2, 24, 6, 30, 6, 30, 22, 6, 2, 30, 24, 90, 6, 33, 42, 59, 6, 40, 30, 72, 22, 77, 6, 73, 2, 66, 30, 66, 24, 72, 90, 60, 6, 99, 66, 122, 42, 48, 118, 144, 6, 432, 40, 60, 30, 132, 72, 66, 22, 129, 154, 870, 6, 210, 146, 264, 2, 60, 66, 224, 30
Offset: 1
Keywords
Examples
For n = 3, c = 2+i: z_0 = 0. z_1 = (z_0)^2 + 2+i = 2+ i (mod 3). z_2 = (z_1)^2 + 2+i = 2+2i (mod 3). z_3 = (z_2)^2 + 2+i = 2 (mod 3). z_4 = (z_3)^2 + 2+i = i (mod 3). z_5 = (z_4)^2 + 2+i = 1+ i (mod 3). z_6 = (z_5)^2 + 2+i = 2 (mod 3) = z_3. So the eventual period is 3, which is the maximum possible over all c and z_0 when n = 3.
Programs
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Maple
f:= proc(N) local pmax,R,S,T,z,a,b,c,x,y,found,zpd,count; pmax:= 0; for a from 0 to N-1 do for b from 0 to N-1 do c:= a+b*I; S:= {}: for x from 0 to N-1 do for y from 0 to N-1 do z:= x+I*y; if not member(z,S) then T:= {z}; R[z]:= 0; found:= false; for count from 1 do z:= expand(z^2 + c) mod N; if member(z,S) then break fi; if member(z,T) then found:= true; zpd:= count - R[z]; break fi; R[z]:= count; T:= T union {z}; od; S:= S union T; if found and zpd > pmax then pmax:= zpd fi; fi; od od; od od; pmax end proc: map(f, [$1..30]); # Robert Israel, Aug 29 2016 -
Mathematica
f[n_] := Module[{pmax = 0, R, S, T, z, a, b, c, x, y, found, zpd, count}, For[a = 0, a <= n - 1, a++, For[b = 0, b <= n - 1, b++, c = a + b*I; S = {}; For[x = 0, x <= n - 1, x++, For[y = 0, y <= n - 1, y++, z = x + y*I; If[!MemberQ[S, z], T = {z}; R[z] = 0; found = False; For[count = 1, True, count++, z = Mod[Expand[z^2 + c], n]; If[MemberQ[S, z], Break[] ]; If [MemberQ[T, z], found = True; zpd = count - R[z]; Break[]]; R[z] = count; T = Union[T, {z}]]; S = Union[S, T]; If [found && zpd > pmax, pmax = zpd]]]]]]; pmax]; Table[f[n], {n, 1, 20}] (* Jean-François Alcover, May 12 2023, after Robert Israel *)
Extensions
More terms from Robert Israel, Aug 29 2016
Comments