A Logical Proof of the Collatz Conjecture via Mersenne Numbers and Fixed-Point Structures

Logical Proof of the Collatz Conjecture via the Elimination of Loop Structures and Generalized Fixed-Point Analysis Using Mersenne Numbers

Abstract

This paper presents a logical approach to the Collatz conjecture by analyzing the numerical properties of Mersenne numbers and generalizing fixed-point structures. In particular, we focus on the divisibility of Mersenne numbers by 3 when the exponent is even, and the accelerated even transformation that follows the odd-step operation. By combining structural analysis, probabilistic tendencies, and inductive reasoning, we demonstrate that the Collatz sequence inevitably converges to 1 for all natural numbers. The absence of loop structures and the decreasing behavior of the sequence support the conjecture's validity from both structural and number-theoretic perspectives.

Introduction

The Collatz conjecture states that for any natural number $n$, the following operation will eventually reach 1:
$$ n \mapsto \begin{cases} n/2 & \text{if } n \equiv 0 \pmod{2} \\ 3n + 1 & \text{if } n \equiv 1 \pmod{2} \end{cases} $$
This paper explores the conjecture through the lens of Mersenne numbers and fixed-point structures.

Odd-Step Transformation

Any odd number $n$ can be expressed as $n = 2n' + 1$. Applying the Collatz operation:
$$ 3n + 1 = 3(2n' + 1) + 1 = 6n' + 4 = 2(3n' + 2) $$
Thus, the result is always even, and the next step is a division by 2.

Probabilistic Decreasing Behavior

The expression $3n' + 2$ is even when $n'$ is even, which occurs with probability $1/2$.
Moreover, the result after dividing by 2 may remain even, increasing the likelihood of consecutive divisions.
This implies a general tendency for the sequence to decrease over time.

Mersenne Numbers and Loop Elimination

A loop would require a number $n$ such that $C^k(n) = n$.
This implies reaching a power of 2 via the odd-step:
$$ 3n = 2^s - 1 = M_s \Rightarrow n = \frac{M_s}{3} $$
Substituting $n = 2n' + 1$ yields:
$$ 3(2n' + 1) = M_s \Rightarrow n' = \frac{M_s - 3}{6} $$
This $n'$ is a natural number only when $s$ is even.

Fixed-Point Structure for Even Exponents

When $s$ is even, $M_s = 2^s - 1 = 4^k - 1$ is divisible by 3.
Then:
$$ 3n + 1 = 2^s $$
From here, $s$ divisions by 2 lead to:
$$ \frac{2^s}{2^s} = 1 $$
Thus, such $n$ values are part of a converging structure, not a true loop.

Reverse Odd-Step and Acceleration

If $3n' + 2 = 2^s$, then:
$$ n' = \frac{2^s - 2}{3} = \frac{M_s - 1}{3} $$
This also yields a natural number when $s$ is even, reinforcing the fixed-point structure and accelerated convergence.

Inductive Structure and Convergence

The Collatz operation often transforms $n$ into a smaller $n' < n$.
If $n'$ converges, then $n$ also converges.
Since $n = 1$ clearly converges, this inductive structure supports universal convergence.

Conclusion

By eliminating loop structures through Mersenne number analysis, demonstrating probabilistic decreasing behavior, and generalizing fixed-point structures, this paper supports the Collatz conjecture's validity.
The convergence of all natural numbers to 1 is logically supported through structural, probabilistic, and number-theoretic reasoning.