Syracuse Conjecture: Algorithmic and Modular Analysis of Increasing and Decreasing Segments

Abstract

This study explores the Syracuse Conjecture, focusing on the effects of its calculation rule: multiply an odd number by 3, add 1, and repeatedly divide by 2 while the result is even. 

The conjecture posits that this process always leads to the number 1. 

Using algorithmic tools, a new structure within Syracuse sequences is revealed, allowing for a systematic definition of increasing and decreasing segments. This segmentation sheds light on the conjecture’s dynamics, particularly the transition patterns between numbers and their successors.

The analysis demonstrates that Syracuse sequences can be decomposed into unordered segments, where the presence and frequency of decreasing segments play a critical role in driving the sequence toward 1. 

A probabilistic framework is introduced, predicting the likelihood of segment behavior based on modular arithmetic. Empirical evidence from over Fifty Syracuse sequences is presented, showing alignment between theoretical and observed frequencies of decreasing segments.

This study concludes that the conjecture can only be refuted by the existence of an infinite, non-decreasing sequence. By leveraging the periodicities and modular patterns inherent to Syracuse sequences, this research provides a foundation for disproving such an occurrence and highlights the pivotal role of structured segments in validating the conjecture. Supporting files with verification data and detailed segment analysis are included.

Link to dataset available here: https://data.mendeley.com/datasets/4hyvzh39b6/5

Keywords: Collatz problem, Syracuse Conjecture, Algorithmic Analysis, Modular Arithmetic