preprints.org > computer science and mathematics > algebra and number theory > doi: 10.20944/preprints202407.0961.v2 (registering DOI)

Preprint Article Version 2 This version is not peer-reviewed

Version 1 : Received: 10 July 2024 / Approved: 11 July 2024 / Online: 11 July 2024 (12:28:15 CEST)
Version 2 : Received: 17 July 2024 / Approved: 18 July 2024 / Online: 18 July 2024 (14:33:11 CEST)

It has been discovered that representing odd integers as modified binary expressions ending with $2^m - 1$ for $m \geq 1$ helps in understanding the dynamics of 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 ending with $2^m - 1$, an even integer that ends in $2^{m+1} + 2^m - 2$ is obtained, which is exactly once divisible by 2, unless the lowest index reduces to zero. This implies that the sequence alternates between odd and even steps $m$ times. This governs the dynamics of the Collatz-type sequences because the value of $m$ determines the number of times the integer can be divided by 2 in each even step. A shortcut method is then presented that gives the even integer after $m$ odd-even steps are completed. This formulation also allows construction of odd integers to follow specific patterns of odd and even steps. The shortcut method for the modified Collatz sequence $5n+1$ is also presented.

Collatz; 3n+1

Computer Science and Mathematics, Algebra and Number Theory

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