preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202407.0961.v1 (registering DOI)

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 10 July 2024 / Approved: 11 July 2024 / Online: 11 July 2024 (12:28:15 CEST)

It is discovered that writing an odd integer as $2^m - 1$ for $m \geq 1$ helps in understanding the dynamics of the Collatz-type sequences. Starting with the original Collatz sequence $3n + 1$, it is found that when the odd step is applied to an odd integer $2^m - 1$, an even integer $2^{m+1} + 2^m - 2$ is obtained, which is exactly once divisible by 2. This implies that the sequence alternates between odd and even steps $m$ times. This governs the dynamics of the Collatz-type sequences because: (i) for the sequence to diverge to infinity, $m$ must be infinite, and (ii) the value of $m$ determines the number of times the integer can be divided by 2 in each even step. This formulation allows construction of odd integers like $\sum 2^m - 1$ to follow specific patterns of odd and even steps. Applying this understanding to the modified Collatz sequence $5n + 1$ reveals that for $m = 3$, the integer $2^{m + 1} + 2^m - 1$ is obtained, indicating that any integer formulation ending with $2^3 - 1$ in $5n + 1$ will diverge to infinity.

Collatz, 3n+1

Computer Science and Mathematics, Algebra and Number Theory

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