Report
Two general and curious conjectures like Collatz's
| العنوان: | Two general and curious conjectures like Collatz's |
|---|---|
| المؤلفون: | Bouhamidi, Abderrahman |
| المساهمون: | Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO) |
| المصدر: | https://hal.science/hal-04559567 ; 2024. |
| بيانات النشر: | HAL CCSD |
| سنة النشر: | 2024 |
| مصطلحات موضوعية: | Collatz conjecture Syracuse Conjecture 3n + 1-problem discrete mathematics, Collatz conjecture, Syracuse Conjecture, 3n + 1-problem, discrete mathematics, [MATH]Mathematics [math] |
| الوصف: | 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. |
| نوع الوثيقة: | report |
| اللغة: | English |
| Relation: | hal-04559567; https://hal.science/hal-04559567; https://hal.science/hal-04559567/document; https://hal.science/hal-04559567/file/Conjectures_Bouhamidi_April_2024.pdf |
| الاتاحة: | https://hal.science/hal-04559567 https://hal.science/hal-04559567/document https://hal.science/hal-04559567/file/Conjectures_Bouhamidi_April_2024.pdf |
| Rights: | http://hal.archives-ouvertes.fr/licences/copyright/ ; info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.F48C4D69 |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |