Conjectures I

Conjecture 1

In base \(2\), for every \(D\), all cycles in \(N_{D}\) have period \(1\).


Conjecture 2

Let us analyze the following figures and tables, depicting the dynamics in base \(2\).

Cycles and precycles of \(f_{4,2}\).

\(D\) \(4\)
Number of cycles \(4\)
Number of preimages of each cycle \(2,4,4,6\)
\(\mbox{Total}=2\,(1+2+2+3)=16=2^{4}\)
Sum (in base 10) of the numbers of preimages of each cycle \(15,30,30,45\)
\(\mbox{Total}=15\,(1+2+2+3)\) \(=15\times 8\) \(=\left(2^{4}-1\right)2^{3}\)

Cycles and precycles of \(f_{6,2}\).

\(D\) \(6\)
Number of cycles \(8\)
Number of preimages of each cycle \(2,\) \(4,\) \(4,\) \(4,\) \(10,\) \(10,\) \(14,\) \(16\)
\(\mbox{Total}=2\,(1+2+2+2+5+5+7+8)\) \(=64\) \(=2^{6}\)
Sum (in base 10) of the numbers of preimages of each cycle \(63,\) \(126,\) \(126,\) \(126,\) \(315,\) \(315,\) \(441,\) \(504\)
\(\mbox{Total}=63\,(1+2+2+2+5+5+7+8)\) \(=63\times 32\) \(=\left(2^{6}-1\right)2^{5}\)

Cycles and precycles of \(f_{8,2}\).

\(D\) \(8\)
Number of cycles \(16\)
Number of preimages of each cycle \(2,\) \(4,\) \(4,\) \(4,\) \(4,\) \(8,\) \(10,\) \(10,\) \(12,\) \(12,\) \(16,\) \(24,\) \(24,\) \(34,\) \(34,\) \(54\)
\(\mbox{Total}=2\,(1+2+2+2+2+4+5+5+6+6+8+12+12+17+17+27)\) \(=256\) \(=2^{8}\)
Sum (in base 10) of the numbers of preimages of each cycle \(255,\) \(510,\) \(510,\) \(510,\) \(510,\) \(1020,\) \(1275,\) \(1275,\) \(1530,\) \(1530,\) \(2040,\) \(3060,\) \(3060,\) \(4335,\) \(4335,\) \(6885\)
\(\mbox{Total}=255\,(1+2+2+2+2+4+5+5+6+6+8+12+12+17+17+27)\) \(=255\times 128\) \(=\left(2^{8}-1\right)2^{7}\)

More generally, in base 2:

In particular, we conclude that the sum of all preimages of the \(2^{\frac{D}{2 }}\) fixed cycles is \[\sum_{i=1}^{m}\left(2^{D}-1\right)p_{i}=\left(2^{D}-1\right)2^{D-1}=\frac{\left(2^{D}-1\right)2^{D}}{2}\] \[= \mbox{sum of all naturals from }1 \mbox{ to }2^{D}-1.\]

Hence, the proof of this conjecture ensures that the first conjecture is also true.

Note: An analogous property, about the number of preimages of the fixed cycles and their respective sum, seems to hold for other bases.