Two general and curious conjectures like Collatz's
Résumé
In this paper, we will introduce two main conjectures. The first one may be seen as a general conjecture that encompasses the Collatz one and may be stated as following.
For any positive integer $p\geq 0$, let $d_p= 2^p+1$, $\alpha_p = 2^p+2$ and $\beta_p = 2^p$.
Starting with any positive integer $n\geq 1$:
If $n$ is divided by $d_p$ then divide it by $d_p$, else multiply it by $\alpha_p$ and add $\beta_p$ times the remainder of $n$ by $d_p$.
Repeating the process iteratively, it always reaches $2^p$ after finite number of iterations. Except for $p=1,3,4$, the process
for $p = 1$ may reaches $2^1$ or also $14$,
for $p = 3$, the process may reaches $2^3$ or also $280$ and
for $p = 4$, the process may reaches $2^4$ or also $1264$. The classical Collatz conjecture may be seen as a special case of our conjecture for $p= 0$ corresponding to $d_0=2$, $\alpha_0=3$ and $\beta_0=1$. The second one, may be stated in its special case as following. Starting with a positive integer:
If it is a multiple of 10 then remove all the zeros on the right, otherwise, multiply it by 6, add 4 times its last digit and divide the result by 5. Repeat the process infinitely. Regardless the starting number, the process eventually reaches 4 after a finite number of iterations. We will discuss the two conjectures more precisely, we will specify the trivial cycles and we will also give a general formulation of the second conjecture. We will discuss verification of both conjectures and give some graphs by specifying the corresponding backward mapping.
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