Investigating Anomalous Behavior in the Collatz Conjecture: Prime Factor Influence and Power Laws

Author(s)	Poorak Gupta, Rajeev Jha
Country	India
Abstract	The Collatz conjecture remains one of the most elusive problems in mathematics. This paper examines the anomalous behavior of certain numbers, particularly those with specific prime factors, in their trajectory towards 1. We introduce a novel hypothesis suggesting that numbers with prime factors such as 3, 7, 19, and 53 exhibit extended stopping times, potentially following a power-law distribution. Through extensive computational analysis, Monte Carlo simulations, and theoretical formulation, we explore the recurrence relations governing these sequences, establishing upper and lower bounds. Our findings provide new insights into the structure of the Collatz sequence and challenge conventional assumptions.
Keywords	Collatz Conjecture , Stopping Time , Prime Factorization , Power Laws , Computational Mathematics , Monte Carlo Simulation , Number Theory , Anomalous Behavior , Recurrence Relations ,Graph Theory
Field	Mathematics
Published In	Volume 6, Issue 6, November-December 2025
Published On	2025-11-06
DOI	https://doi.org/10.63363/aijfr.2025.v06i06.1880
Short DOI	https://doi.org/g99qps

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