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An Extension of the Collatz Conjecture modulo $2^p+2^q$

Abderrahman Bouhamidi

TL;DR

Extends the Collatz problem by introducing a generalized operator $T$ parameterized by $(d,\alpha,\beta)_{\pm}$ that unifies the classical map and its modular variants. The main construction specializes to families with $d=2^p+2^q$, including the modulo $10$ case $(10,12,8)_+$, and yields explicit forward-iteration and cycle-structure results. The paper provides an admissibility framework, a total stopping time theory, and a backward-mapping approach with inverse graphs, complemented by substantial computational verification (up to $p\le25$ and $n\le10^7$, and $n\le6.5\times10^9$ for a key case). Together, these elements offer a scalable framework for analyzing generalized Collatz-like maps, enabling both theoretical analysis and extensive computational testing, and paving the way for broader modular extensions.

Abstract

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo $10$ which may be stated as follows. Starting from any positive integer, if it is a multiple of $10$ then divide it by 10, otherwise, multiply it by $12$, add $8$ times its last digit and divide the result by $10$. Repeat the process infinitely. Regardless the starting number, the process eventually reaches $4$ after a finite number of iterations. The genaral conjecture studied here will encompasse the classical Collatz conjecture togher with our proposed one modulo $10$.

An Extension of the Collatz Conjecture modulo $2^p+2^q$

TL;DR

Extends the Collatz problem by introducing a generalized operator parameterized by that unifies the classical map and its modular variants. The main construction specializes to families with , including the modulo case , and yields explicit forward-iteration and cycle-structure results. The paper provides an admissibility framework, a total stopping time theory, and a backward-mapping approach with inverse graphs, complemented by substantial computational verification (up to and , and for a key case). Together, these elements offer a scalable framework for analyzing generalized Collatz-like maps, enabling both theoretical analysis and extensive computational testing, and paving the way for broader modular extensions.

Abstract

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo which may be stated as follows. Starting from any positive integer, if it is a multiple of then divide it by 10, otherwise, multiply it by , add times its last digit and divide the result by . Repeat the process infinitely. Regardless the starting number, the process eventually reaches after a finite number of iterations. The genaral conjecture studied here will encompasse the classical Collatz conjecture togher with our proposed one modulo .
Paper Structure (10 sections, 4 theorems, 37 equations, 4 figures, 1 table, 2 algorithms)

This paper contains 10 sections, 4 theorems, 37 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. The $(2^p+2^{q+1})n+2^p$ problem modulo $2^p+2^q$ where $p\geq q\geq0$ as an extension of the Collatz conjecture
  3. Case $d=2^qd'\geq 2$ with $d'\geq 2$ and $q\geq 0$
  4. Particular case where $d=2^p+2^q$ with $p\geq q\geq 0$
  5. Particular case where $p=q\geq 0$
  6. Some iterate and total stopping time formulas
  7. Computation and verification
  8. Verification of the main conjecture
  9. Backward mapping and graphs
  10. Conclusion

Key Result

Theorem 2.1

Let $q\geq 0$, for $d=2^qd'\geq 2$, then the triplet $(d,\alpha,\beta)_{\pmb{+}}:=(d,d+2^q,d-2^q)_{\pmb{+}}$ is admissible and it has at least the following trivial cycle of length $\dfrac{1}{2^q}d+q$: for $q\geq 1$. For $q=0$ the trivial cycle is reduced to the following cycle of length $d$:

Figures (4)

  • Figure 1: Trajectories : Strating from $n=75$ (left) and from $n=135$ (right)
  • Figure 3: Tree Graph with its unique cycle corresponding to the triplet $(2,3,1)_+$ (left) and to triplet $(10,12,8)_+$.
  • Figure 4: Tree Graph with its unique cycle corresponding to the triplet $(5,6,4)_+$ (left) and to triplet $(8,12,4)_+$ with its two cycles.
  • Figure 5: Tree Graph with its unique cycle corresponding to the triplet $(128,192,64)_+$ (left) and to triplet $(8192,12288,4096)_+$.

Theorems & Definitions (17)

  • Conjecture 1.1
  • Definition 1.1
  • Conjecture 1.2: of Collatz
  • Conjecture 1.3
  • Theorem 2.1
  • proof
  • Example 2.1
  • Conjecture 2.1
  • Example 2.2
  • Conjecture 2.2
  • ...and 7 more