The dynamics of a trick 
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 \(2\).
- \(D=4\)
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 | \(15,30,30,45\) \(\mbox{Total}=15\,(1+2+2+3)\) \(=15\times 8\) \(=\left(2^{4}-1\right)2^{3}\) |
- \(D=6\)
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 | \(63,\) \(126,\) \(126,\) \(126,\) \(315,\) \(315,\) \(441,\) \(504\) \(\mbox{Total}=63\,(1+2+2+2+5+5+7+8)\) \(=63\times 32\) \(=\left(2^{6}-1\right)2^{5}\) |
- \(D=8\)
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:
- there is one and only one (fixed) cycle with two pre-images, \(F_{1} = \{0101...01\};\) notice that the sum (in base 10) of the numbers of the preimages of this cycle is \(2^{D} - 1;\)
- in the remaining \(2^{\frac{D}{2}} - 1\) cycles, which we denote by \(F_{2},...,F_{m}\), where \(m=2^{\frac{D}{2}}-1\), each with \(2p_{i}\) preimages (an even number because, with the exception of the natural numbers which are equal to their reverse and of which we know there is an even number, a number and its reverse belong to the preimage of the same cycle) and whose numbers have sum (in base \(10\)) equal to \(S_{i}\), one has \[S_{i} = \left(2^{D} - 1\right)p_{i}\] and \[\sum_{i=1}^{m}p_{i}=2^{D-1}.\]
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.
