TL;DR
This paper introduces new theorems and algorithms that deepen understanding of the Collatz conjecture by analyzing iteration counts, peak values, and relationships between starting numbers and their trajectories.
Contribution
It presents novel theorems, corollaries, and algorithms that explore properties and relationships related to the Collatz conjecture, advancing theoretical insights.
Findings
New theorems relating to Collatz iteration counts
Algorithms for analyzing number trajectories in Collatz sequences
Insights into relationships between starting numbers and peak values
Abstract
Proposed in 1937, the Collatz conjecture has remained in the spotlight for mathematicians and computer scientists alike due to its simple proposal, yet intractable proof. In this paper, we propose several novel theorems, corollaries, and algorithms that explore relationships and properties between the natural numbers, their peak values, and the conjecture. These contributions primarily analyze the number of Collatz iterations it takes for a given integer to reach 1 or a number less than itself, or the relationship between a starting number and its peak value.
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