Discrete Dynamical Systems Modulo $2^p+2^q$ as an Extension to the Collatz Conjecture. (Conjectured by A. Bouhamidi)

Notations and Preliminaries

Let $\mathbb{N} = \{1, 2, \dots\}$ denote the set of positive integers. For any integers $p \ge q \ge 0$, we consider the parameters

$$d = 2^p + 2^q, \quad \alpha = 2^p + 2^{q+1}, \quad \text{and} \quad \beta = 2^p,$$

and the map $T_{p,q} : \mathbb{N} \longrightarrow \mathbb{N}$ defined by [1] and [2]:

$$T_{p,q}(n) = \begin{cases} n/d & \text{if } n \equiv 0 \pmod d, \\ \dfrac{\alpha n + \beta [n]_d}{d} & \text{if } n \not\equiv 0 \pmod d, \end{cases}$$

where $[n]_d \in \{0, 1, \dots, d-1\}$ denotes the remainder in the Euclidean division of $n$ by $d$. We refer interchangeably to the triplet $(2^p+2^q, 2^p+2^{q+1}, 2^p)$ or to the map $T_{p,q}$.

A sequence $\Omega(\omega) = \left(\omega \to T_{p,q}(\omega) \to T_{p,q}^{(2)}(\omega) \to \dots \to T_{p,q}^{(k)}(\omega) = \omega\right)$ denotes a cycle of length $k$ starting from its smallest element $\omega$.

As shown in [1], [2] (see references below), for all $p \ge q \ge 0$, the map $T_{p,q}$ possesses at least the trivial cycle $\Omega(2^{p-q})$ of length $2^{p-q} + q + 1$, then:

$$T_{p,q}^{(2^{p-q}+q+1)}(2^{p-q}) = 2^{p-q}.$$

We define the exceptional set $\mathbb{E}$ as:

$$\mathbb{E} = \{(1,0), (2,1), (2,2), (3,0), (4,0), (5,2), (6,2), (7,0)\}.$$

Main Conjecture (Bouhamidi, 2026): For all integers $p \ge q \ge 0$, the dynamic behavior of the map $T_{p,q}$ satisfies [A], [B] and [C] given below:

[A] Order One Case: If $(p,q) \notin \mathbb{E}$, the map $T_{p,q}$ has the trivial cycle $\Omega(2^{p-q})$ as its unique cycle. That is,

$$\forall n \ge 1, \; \exists k \ge 0 : T_{p,q}^{(k)}(n) = 2^{p-q}.$$

[B] Order Two Case: If $(p,q) \in \mathbb{E}$ and $(p,q) \neq (5,2)$, the map $T_{p,q}$ has exactly two cycles: the trivial cycle $\Omega(2^{p-q})$ and a unique secondary cycle $\Omega(\omega_{p,q})$. That is,

$$\forall n \ge 1, \; \exists k \ge 0 : T_{p,q}^{(k)}(n) \in \{2^{p-q}, \omega_{p,q}\}.$$

[C] Order Three Case: If $(p,q) = (5,2)$, the map $T_{5,2}$ associated with the triplet $(36, 40, 32)$ has exactly three cycles: the trivial cycle $\Omega(2^{5-2}=8)$, and two secondary cycles $\Omega(\omega_{5,2}^{(1)} = 76200)$ of length $70$ and $\Omega(\omega_{5,2}^{(2)} = 87176)$ of length $35$. That is,

$$\forall n \ge 1, \; \exists k \ge 0 : T_{5,2}^{(k)}(n) \in \{8, 76200, 87176\}.$$

Secondary Cycles for the Exceptional Cases in $\mathbb{E}$

For completeness, the explicit secondary cycles associated with each pair $(p,q) \in \mathbb{E}$ are detailed below: The cycles $\Omega(\omega_{1,0})$, $\Omega(\omega_{2,1})$, $\Omega(\omega_{2,2})$, $\Omega(\omega_{3,0})$, $\Omega(\omega_{4,0})$, $\Omega(\omega^{(1)}_{5,2})$, $\Omega(\omega^{(2)}_{5,2})$, $\Omega(\omega_{6,2})$ and $\Omega(\omega_{7,0})$ starting by $\omega_{1,0}=14$, $\omega_{2,1}=74$, $\omega_{2,2}=67$, $\omega_{3,0}=280$, $\omega_{4,0}=1264$, $\omega^{(1)}_{5,2}=76200$, $\omega^{(2)}_{5,2}=87176$, $\omega_{6,2}=1264$ and $\omega_{7,0}=3027584$, respectively and of length $9$, $7$, $6$, $21$, $49$, $70$, $35$, $69$, $630$ respectively are given as follows:

$$\begin{array}{ll} \Omega(\omega_{1,0})=& \bigl(14\rightarrow 20\rightarrow 28\rightarrow 38\rightarrow 52\rightarrow 70\rightarrow 94 \rightarrow 126\rightarrow 42\rightarrow 14\bigr),\\ \Omega(\omega_{2,1})=& \bigl(74\rightarrow 100\rightarrow 136\rightarrow 184\rightarrow 248\rightarrow 332\rightarrow 444 \rightarrow 74 \bigr),\\ \Omega(\omega_{2,2})=& \bigl(67\rightarrow102\rightarrow156\rightarrow236\rightarrow356\rightarrow536\rightarrow67\bigr),\\ \Omega(\omega_{3,0})=& \bigl(280\rightarrow 312\rightarrow 352\rightarrow 392\rightarrow 440\rightarrow 496\rightarrow 552\rightarrow 616\rightarrow 688\rightarrow 768\rightarrow856\rightarrow952\rightarrow\\ & 1064\rightarrow 1184\rightarrow 1320\rightarrow 1472\rightarrow 1640\rightarrow 1824\rightarrow 2032\rightarrow 2264 \rightarrow 2520 \rightarrow280),\\ \Omega(\omega_{4,0})=&\bigl( 1264\rightarrow 1344\rightarrow 1424\rightarrow 1520\rightarrow 1616\rightarrow 1712\rightarrow 1824\rightarrow 1936\rightarrow 2064\rightarrow\\ &2192\rightarrow 2336\rightarrow 2480\rightarrow 2640\rightarrow 2800\rightarrow 2976 \rightarrow 3152\rightarrow3344\rightarrow 3552\rightarrow 3776\\ &\rightarrow 4000\rightarrow4240\rightarrow 4496\rightarrow 4768\rightarrow 5056\rightarrow 5360\rightarrow 5680\rightarrow 6016\rightarrow 6384\\ &\rightarrow6768\rightarrow 7168 \rightarrow 7600\rightarrow 8048\rightarrow 8528\rightarrow 9040\rightarrow 9584\rightarrow 10160\rightarrow 10768 \\ & \rightarrow 11408\rightarrow 12080\rightarrow 12800\rightarrow 13568\rightarrow 14368\rightarrow 15216\rightarrow 16112\rightarrow 17072\rightarrow \\ & 18080\rightarrow 19152\rightarrow 20288\rightarrow 21488\rightarrow 1264\bigr),\\ \Omega(\omega^{(1)}_{5,2})=& \bigl(76200\rightarrow84688\rightarrow94112\rightarrow104576\rightarrow116224\rightarrow129152\rightarrow143520\rightarrow159488\\ &\rightarrow177216\rightarrow196928\rightarrow218816\rightarrow243136\rightarrow270176\rightarrow300224\rightarrow333600\rightarrow\\&370688\rightarrow411904\rightarrow457696\rightarrow508576\rightarrow565088\rightarrow627904\rightarrow697696\rightarrow775232\\&\rightarrow861376\rightarrow957088\rightarrow1063456\rightarrow1181632\rightarrow1312928\rightarrow1458816\rightarrow1620928\\&\rightarrow1801056\rightarrow2001184\rightarrow2223552\rightarrow2470624\rightarrow2745152\rightarrow3050176\rightarrow3389088\\&\rightarrow3765664\rightarrow4184096\rightarrow4649024\rightarrow5165600\rightarrow5739584\rightarrow6377344\rightarrow7085952\\&\rightarrow196832\rightarrow218720\rightarrow243040\rightarrow270048\rightarrow300064\rightarrow333408\rightarrow370464\rightarrow\\&411648\rightarrow457408\rightarrow508256\rightarrow564736\rightarrow627488\rightarrow697216\rightarrow774688\rightarrow860768\\&\rightarrow956416\rightarrow1062688\rightarrow1180768\rightarrow1311968\rightarrow1457760\rightarrow1619744\rightarrow1799744\\&\rightarrow1999744\rightarrow2221952\rightarrow2468864\rightarrow2743200\rightarrow76200\bigr),\\ \end{array}$$ $$\begin{array}{ll} \Omega(\omega^{(2)}_{5,2})=& \bigl(87176\rightarrow96880\rightarrow107648\rightarrow119616\rightarrow132928\rightarrow147712\rightarrow164128\rightarrow182368\\& \rightarrow202656\rightarrow225184\rightarrow250208\rightarrow278016\rightarrow308928\rightarrow343264\rightarrow381408\\&\rightarrow423808\rightarrow470912\rightarrow523264\rightarrow581408\rightarrow646016\rightarrow717824\rightarrow797600\\&\rightarrow886240\rightarrow984736\rightarrow1094176\rightarrow1215776\rightarrow1350880\rightarrow1500992\rightarrow1667776\\&\rightarrow1853088\rightarrow2059008\rightarrow2287808\rightarrow2542016\rightarrow2824480\rightarrow3138336\rightarrow87176\bigr),\\ \Omega(\omega_{6,2})=&\bigl(1264\rightarrow1376\rightarrow1472\rightarrow1600\rightarrow1728\rightarrow1856\rightarrow1984\rightarrow2112\rightarrow2240\rightarrow2432\\ &\rightarrow2624\rightarrow2816\rightarrow3008\rightarrow3200\rightarrow3392\rightarrow3648\rightarrow3904\rightarrow4160\rightarrow4416\rightarrow4736\\&\rightarrow5056\rightarrow5376\rightarrow5696\rightarrow6080\rightarrow6464\rightarrow6848\rightarrow7296\rightarrow7744\rightarrow8256\rightarrow8768\\&\rightarrow9344\rightarrow9920\rightarrow10560\rightarrow11200\rightarrow11904\rightarrow12608\rightarrow13376\rightarrow14208\rightarrow15104\\&\rightarrow16000\rightarrow16960\rightarrow17984\rightarrow19072\rightarrow20224\rightarrow21440\rightarrow22720\rightarrow24064\rightarrow25536\\&\rightarrow27072\rightarrow28672\rightarrow30400\rightarrow32192\rightarrow34112\rightarrow36160\rightarrow38336\rightarrow40640\\&\rightarrow43072\rightarrow45632\rightarrow48320\rightarrow51200\rightarrow54272\rightarrow57472\rightarrow60864\rightarrow64448\rightarrow\\&68288\rightarrow72320\rightarrow76608\rightarrow81152\rightarrow85952\rightarrow1264\bigr),\\ \Omega(\omega_{7,0})=&\bigl( 3027584\rightarrow3051136\rightarrow3074816\rightarrow3098752\rightarrow 3122816\rightarrow3147136\rightarrow3171584\rightarrow\\&3196288\rightarrow3221120\rightarrow3246208\rightarrow3271424\rightarrow3296896\rightarrow3322496\rightarrow3348352\rightarrow\ldots\ldots\\& \ldots\rightarrow 367160576\rightarrow370006784\rightarrow372875136\rightarrow375765760\rightarrow378678784\rightarrow381614336\rightarrow\\& 384572672\rightarrow387553920\rightarrow390558336\rightarrow3027584\bigr). \end{array}$$

$(p,q)$	Triplet $(d,\alpha,\beta)$	Trivial Cycle $\Omega(2^{p-q})$	Secondary Cycle Base $\omega_{p,q}$	Length $k$
$(1,0)$	$(3,4,2)$	$\Omega(2)$	$\omega_{1,0} = 14$	9
$(2,1)$	$(6,8,4)$	$\Omega(2)$	$\omega_{2,1} = 74$	7
$(2,2)$	$(8,12,4)$	$\Omega(1)$	$\omega_{2,2} = 67$	6
$(3,0)$	$(9,10,8)$	$\Omega(8)$	$\omega_{3,0} = 280$	21
$(4,0)$	$(17,18,16)$	$\Omega(16)$	$\omega_{4,0} = 1264$	49
$(5,2)$	$(36,40,32)$	$\Omega(8)$	$\omega_{5,2}^{(1)} = 76200$	70
			$\omega_{5,2}^{(2)} = 87176$	35
$(6,2)$	$(68,72,64)$	$\Omega(16)$	$\omega_{6,2} = 1264$	69
$(7,0)$	$(129,130,128)$	$\Omega(128)$	$\omega_{7,0} = 3027584$	630

Table: Explicit secondary cycles for all parameters $(p,q) \in E$.

Particular Cases

1. Classical Collatz Conjecture ($p=q=0$): Yields the accelerated Collatz map with trivial cycle $\Omega(1) = (1 \to 2 \to 1)$ of length $L = 2$.

2. Modulo-10 Collatz-like Conjecture ($p=3, q=1$): Gives $d = 10$.

Choose any integer $n \ge 1$. If $n$ ends in $0$, divide it by $10$. Otherwise, multiply $n$ by $12$, add $8$ times the last digit of $n$, and divide the result by $10$. Repeat the process iteratively.

The conjecture states that, for any choice of the integer $n \ge 1$, the process eventually reaches $4$ and enters the trivial cycle:

$$\Omega(4) = (4 \to 8 \to 16 \to 24 \to 32 \to 40 \to 4),$$

of length $L = 6$.

3. Diagonal Case ($p=q$): Associated with $(2^{p+1}, 2^p\times 3, 2^p)$, i.e. the classical Collatz triplet $(2,3,1)$ scaled by $2^p$:

$$(2^{p+1}, 2^p\times 3, 2^p)=(2^{p}\times 2, 2^p\times 3, 2^p\times 1)=2^p\times (2,3,1).$$

For all $p \neq 2$, $T_{p,p}$ has the unique trivial cycle $\Omega(1)$ of length $L = p + 2$. The trivial cycle of length $p+2$ is

$$\Omega(1) = (1 \to 2 \to 2^2 \to 2^3 \to \ldots \to 2^{p+1} \to 1),$$

For $p=2$, we have $(2,2) \in \mathbb{E}$ with two cycles: $\Omega(1)$ and $\Omega(67)$.

4. Connection to Fermat Numbers ($q=0$): Here $d = 2^p + 1$. When $p = 2^k$, $d$ is a Fermat number. The case $(4,0) \in \mathbb{E}$ yields two cycles: $\Omega(16)$ and $\Omega(1264)$.

Related MathDB Problems

The present conjecture is part of a broader study of Collatz-type maps and admissible triplets. In particular:

• MathDB 382462 and 372483— Diagonal case $p=q$ of the present conjecture, corresponding to

$$(2^p\times2,\,2^p\times3,\,2^p\times1) =2^p(2,3,1).$$

Related particular case of the present conjecture.

• MathDB 372482— Modulo-10 case $p=3,q=1$ of the present conjecture, corresponding to $(10,12,8)$.

The above problems are based on the research framework developed in [1].

• MathDB 382461 — Related admissibility conjecture for Mersenne-type triplets

$$(2^p-1,2^p-1,1)_+.$$

This problem is based on the research framework developed in [2].