MONOTONICITY AND CONVERGENCE IN THE
COLLATZ CONJECTURE: A NEW
PERSPECTIVE
MONOTONICIDAD Y CONVERGENCIA EN LA
CONJETURA DE COLLATZ: UNA NUEVA PERSPECTIVA
Guillermo Wells Abascal
TecSalud, Tecnológico de Monterrey - México
Angel Gabriel Zavala Raus
TecSalud, Tecnológico de Monterrey - México
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DOI: https://doi.org/10.37811/cl_rcm.v8i6.15985
Monotonicity and Convergence in the Collatz Conjecture: A New
Perspective
Guillermo Wells Abascal1
guillermowells@live.com
https://orcid.org/0009-0009-1180-4147
TecSalud, Tecnológico de Monterrey
México
Angel Gabriel Zavala Raus
angelgabrielzavalaraus@hotmail.com
https://orcid.org/0009-0007-2253-4678
TecSalud, Tecnológico de Monterrey
México
ABSTRACT
The Collatz conjecture declares that every positive integer will eventually reach 1 when subjected to a
simple iterative process: if the number is even, it is divided by 2, and if it is odd, it is multiplied by 3
and then increased by 1. Despite the straightforward nature of these rules, a general proof of the
conjecture remains elusive. For the above, this study introduces an alternative interpretation of the
conjecture. This approach involves multiplying an odd integer by 3 and subsequently adding the
largest power-of-2 factor within . Repeated iterations of this alternative process show that any initial
odd integer will eventually convert into a power of 2, leading the sequence towards convergence.
The behavior of the sequence was studied by representing the resulting integers as a power of 2
multiplied by an odd component. Using this representation under the modified rules, we developed a
structured proof framework that demonstrates the consistent reduction of the odd component’s relative
value after each iteration, the accelerated increase of the power-of-2 factor’s relative value, and the
absence of any divergent cycles or alternative behaviors. This analysis provides insights into the
mechanics of convergence in the Collatz sequence and proposes a new perspective for understanding
the conjecture’s underlying dynamics.
Keywords: Collatz conjecture, convergence, iterative sequences, number theory
1
Autor Principal
Correspondencia: guillermowells@live.com
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Monotonicidad y Convergencia en la Conjetura de Collatz: Una Nueva
Perspectiva
RESUMEN
La conjetura de Collatz establece que todo entero positivo llegará a ser 1 si es sometido a un proceso
iterativo simple: si el número es par, se divide entre 2, y si es impar, se multiplica por 3 y luego se suma
1. A pesar de la naturaleza sencilla de estas reglas, no se ha podido demostrar de manera general la
conjetura. Por lo anterior, este estudio presenta una interpretación alternativa de la conjetura. Este
enfoque implica multiplicar un entero impar N1 por 3 y, posteriormente, sumar el factor-potencia de 2
más grande de N1. Al repetir este proceso iterativo, cualquier entero impar inicial N1 se convertirá
eventualmente en una potencia de 2, lo que lleva la secuencia hacia la convergencia. Se estudió el
comportamiento de la secuencia representando los números enteros resultantes como una potencia de 2
multiplicada por un componente impar. Utilizando esta representación bajo las reglas modificadas,
desarrollamos un procedimiento que demuestra la reducción consistente del valor relativo del
componente impar después de cada iteración, el aumento acelerado del valor relativo del factor-potencia
de 2 y la ausencia de ciclos divergentes o comportamientos alternativos. Este análisis proporciona
información sobre la mecánica de la convergencia en la secuencia de Collatz y propone una nueva
perspectiva para comprender el comportamiento subyacente de la conjetura.
Palabras clave: conjetura de Collatz, convergencia, secuencias iterativas, teoría de números
Artículo recibido 17 octubre 2024
Aceptado para publicación: 25 noviembre 2024
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if is odd
if is even
INTRODUCTION
The Collatz conjecture explores the behavior of a sequence initiated by any positive integer, where each
subsequent term is generated according to the rules given in Eq. (1). The conjecture posits that this
series always reaches 1 regardless of the chosen initial integer [7, 15].
……………...…. ……. ……nt+1
(1)
For any positive integer , the Collatz sequence proceeds as follows: if is even, the next term is
obtained by dividing it by 2; if is odd, it is transformed by multiplying it by 3 and adding 1 [14]. The
conjecture also implies that the followed sequence will always arrive to the trivial cycle “1-4-2-1” [2].
Originally proposed by Lothar Collatz in 1937, the Collatz conjecture is also known by several other
names, including the conjecture, the Ulam conjecture, Kakutani’s problem, the Thwaites
conjecture, Hasse’s algorithm, the Syracuse problem or the hailstone sequence [1]. Although a formal
proof of the conjecture remains elusive, extensive experimental evidence and heuristic arguments
suggest that it is valid [5], and the conjecture has been verified for values up to [6]. In
addition, one of the most notable recent advances was made by Terence Tao, who proved that most
orbits of the Collatz map attain almost bounded values [8, 13]. Finally, proving the Collatz conjecture
is equivalent to demonstrating the absence of cycles other than “1-4-2-1” and of divergent orbits [10].
Bottom-up Approach to the Collatz Conjecture
A bottom-up representation of the Collatz conjecture suggests that any positive integer can be reached
by applying an inverse form of the rules described in Eq. (1), these being and
[12]. Starting
from , both rules can be applied as long as only integers are produced.
Table 1 illustrates the steps required to reach the first 10 integers, showing a complex behavior [3].
However, to prove the validity of the Collatz conjecture it suffices to prove that it holds true for every
positive odd integer [9]. This occurs because each odd number can generate multiples of itself in the
form , demonstrating that every even number originates from an odd number. Accordingly, the
formulas in this study are designed to generate only positive odd numbers.
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Table 1 Steps required to generate the first 10 positive integers nt following the rules of the Collatz
conjecture.
nt
Steps to Reach nt
1
1
2
1, 2
3
1, 2, 4, 8, 16, 5, 10, 3
4
1, 2, 4
5
1, 2, 4, 8, 16, 5
6
1, 2, 4, 8, 16, 5, 10, 3, 6
7
1, 2, 4, 8, 16, 5, 10, 20, 40, 13, 26, 52, 17, 34, 11, 22, 7
8
1, 2, 4, 8
9
1, 2, 4, 8, 16, 5, 10, 20, 40, 13, 26, 52, 17, 34, 11, 22, 7, 14, 28, 9
10
1, 2, 4, 8, 16, 5, 10
Progression of the Bottom-Up Collatz Conjecture: Examples and Diagram
Fig. 1 illustrates the initial integers that can be generated by starting from and applying the rules
of the bottom-up Collatz conjecture. The diagram indicates that the
formula cannot be applied to
all numbers or “branches”, but it consistently generates odd numbers when used. As an example, the
equations in the figure detail the steps taken to reach the numbers 5 and 3.
Figura 1. a) Representation of the first integers that can be generated by starting from n1 = 1 and
following both rules of the bottom-up Collatz conjecture. Black arrows indicate the doubling of the
current number (2n_t), while red arrows represent the use of the (n_t-1)/3 formula. b) Algebraic
expressions that indicate the steps required to reach number 5. c) Algebraic expressions that indicate
the steps needed to reach number 3.
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General Formula to Represent the Steps Required to Generate Any Odd Number nt
The reverse Collatz iterations used to reach an odd integer can be represented as functions [11]. As
illustrated in Fig. 1, the steps needed to reach any odd number nt starting from can be represented
by the following general formula:
,
where: nt, , ,; . Here, is the total number of times the
transformation 2nt was applied, while is the total number of times the
formula was applied.
Eq. (2) captures the iterative process of the bottom-up Collatz sequence to generate any odd integer .
The exponents denote the cumulative number of times that was multiplied by 2 between
successive applications of the
formula. For example, the first step to generate 17 from 1 = 1
involves doubling four times before applying
for the first time, which corresponds to .
Then, between the first and second applications of
, is doubled three times, so − = 3, and so
on.
Alternative Interpretation of the Collatz Conjecture
The terms of Eq. (2) can be rearranged to yield Eq. (3), in which is an initial odd positive integer.
where: N1, , ,; . Here, represents the total number of times
the transformation 2nt was applied, while β is the total number of times the
formula was applied.
Transformation of Odd Positive Integers N1 Into Powers of 2 Via Iteratively Multiplying by 3 and
Adding a Power-of-2 Factor
Based on Eq. (3), an odd positive integer can be transformed into a power of 2 by applying the
following iterative steps:
• Multiply by 3 and add 1 (20) to obtain an even integer .
• For each subsequent iteration, multiply by 3 and add the largest power of 2 that is a factor of .
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• Repeat this process until transforms into a power of 2, denoted .
The iterative process is represented by Eq. (4):
where is the biggest power of 2 that divides Nt.
As an example, Table 2 summarizes the steps that must be followed to convert odd positive integers ,
from 3 to 13, into powers of 2. Each positive integer is expressed as a power of 2 multiplied by an
odd number.
Table 2. Summary of the transformation of positive odd integers from 3 to 13 by multiplying each
by 3 and adding the largest power of 2 that divides it (including 20). This cycle is repeated until the
integer becomes a power of 2. Each resulting integer is represented as a power of 2 multiplied by an
odd factor.
N1
3N1 + 1
3N2 + 2a
3N3 + 2b
3N4 + 2c
3
19+1 = 215 →
2115+21 = 25
5
115+1 = 24
7
121+1 = 2111 →
2133+21 = 2217 →
2251+22 = 2413 →
2439+24 = 275 → 211
9
127+1 = 227 →
2221+22 = 2311 →
2333+23 = 2417 →
2451+24 = 2613 → 295 → 213
11
133+1 = 2117 →
2151+21 = 2313 →
2339+23 = 265 →
2615+26 = 210
13
139+1 = 235 →
2315+23 = 27
Note: For N1 = 7 and N1 = 9, the final steps were simplified.
Monotonic Reduction and Convergence of the Alternative Interpretation of the Collatz
Conjecture
Eq. (4) presents a unique scenario that cannot be replicated by iteratively multiplying by larger odd
factors at the start of each cycle (e.g., , , etc.). This behavior arises because the
rules of the Collatz conjecture allow the power-of-2 factor of Nt to increase with each cycle, including
a growth in the power-of-2 factor’s relative value.
The following sections outline the key properties that explain why any initial positive odd integer N1
can ultimately be transformed into a power of 2 () by applying the steps of this alternative
interpretation of the Collatz conjecture.
Lemma 4.1 Iteratively adding the largest power-of-2 factor of an odd positive integer will ultimately
transform into the closest power of 2 that is greater than N1.
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Proof Every positive integer Nt can be uniquely factorized into a power of 2 multiplied by an odd
component:
Ot,
where is the odd component of , and is the largest power-of-2 factor of .
Similarly, every positive integer Nt has a unique binary representation [4]:
,
where 0 ≤ a1 < a2 < … < an.
The smallest term of the binary representation, , is equal to the biggest power-of-2 factor, .
This can be proven if the previous formula is rewritten to resemble a power-of-2 factor multiplied by
an odd component:
,
where the term in parentheses is equal to the odd component Ot.
After one iteration, upon summing (or equivalently ) to Nt, doubles:
After sufficient iterations, the initial becomes equal to , and their sum produces . This
process continues, systematically combining and doubling terms until all binary terms coalesce
into, the closest power of 2 that is greater than the positive odd integer N1.For a detailed example
of this iterative process, refer to Table 3.
Table 3. Test to analyze the effect over positive odd integers N1 from 3 to 15 (decomposed into series
of sums of non-repeating powers of 2), after adding the smallest power of 2 found in their respective
series. The cycle is repeated until the number becomes a power of 2.
N1
N1 + 1
N2 + 2a
N3 + 2b
3 (1+2)
1+1+2 = 4
5 (1+4)
1+1+4 = 2+4 →
2+2+4 = 8
7 (1+2+4)
1+1+2+4 = 8
9 (1+8)
1+1+8 = 2+8 →
2+2+8 = 4+8 →
4+4+8 = 16
11 (1+2+8)
1+1+2+8 = 4+8 →
4+4+8 = 16
13 (1+4+8)
1+1+4+8 = 2+4+8 →
2+2+4+8 = 16
15 (1+2+4+8)
1+1+2+4+8 = 16
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Conclusion
The iterative summation of the largest power-of-2 factor of the odd positive integer N1 successively
eliminates all smaller binary terms of N1 by combining and doubling them, ultimately transforming the
integer into the nearest power of 2 larger than N1.
Lemma 4.2 Let denote an odd positive integer undergoing the iterative transformation 3Nt
+ , with being the largest power-of-2 factor in . ’s odd factor divided by its largest power-
of-2 factor, undergoes a strict monotonic reduction in each cycle.
Proof Representing each number as a power of 2 multiplied by an odd component:
Ot,
where is the odd component of and is the biggest power-of-2 factor of . We seek to show
that the value
decreases in every iteration, demonstrating a monotonic reduction of the relative
value of in the sequence.
Worst-Case Scenario Let be a hypothetical positive odd integer that undergoes the iterative
transformation . Assume that in this process only consecutive powers of 2 (i.e. 1, 2,
4, 8, …) are added in each iteration. Under these conditions, this sequence represents a worst-case
scenario in which the sequence progresses at its slowest rate. Even so, it can be shown that this sequence
progresses monotonically, with the odd component decreasing relative to its corresponding power of 2
at each step.
, , , , …
,
,
, …
Dividing the odd components by their powers of 2, we obtain:
,
,
,
, …
When N1 > 1, the terms of the series always decrease more rapidly after every new iteration.
Non-Worst-Case Scenario For any iteration of the process applied to the
hypothetical positive odd integer N1*, if in any step the added power-of-2 factor () does not follow
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the minimal consecutive sequence, the ratio of the odd component Ot* to the power of 2 will decrease
even more rapidly than in the worst-case scenario:
where j is a positive integer.
CONCLUSION
The relative value of the odd component Ot decreases with each iteration, as shown by the construction
of the worst-case scenario sequence. Even in this scenario, the sequence progresses monotonically, with
the odd component’s influence diminishing over time as the relative value of the power of 2 factor
increases. Also, any variation in the sum of powers of 2 that includes major factors accelerates the
process of turning Nt to a power of 2. These cases eliminate the possibility of strange cycles or sequence
divergence, supporting the hypothesis that any positive odd integer N1 approaches a power of 2 through
repeated iterations under the Collatz process.
Author Contributions
Conceptualization: Wells Abascal G.; Methodology: Wells Abascal G.; Formal analysis and
investigation: Wells Abascal G.; Writing - original draft preparation: Wells Abascal G.; Writing -
review and editing: Wells Abascal G., Zavala Raus A.
Funding No funding was received for conducting this study.
Declarations
Conflict of interest We have nothing to declare and there is no conflict of interest.
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