TL;DR
This paper investigates the counting function for numbers satisfying the Collatz conjecture, introducing a method to construct solutions to a related exponential congruence and estimating the number of Collatz numbers up to x.
Contribution
It presents a novel approach to construct solutions to an exponential congruence related to the Collatz problem, achieving a lower bound of at least x^{0.3227} for the count of Collatz numbers.
Findings
Constructed solutions from free variables for the exponential congruence.
Established a lower bound of x^{0.3227} for Collatz numbers in [1,x].
Compared with the historical record of 0.84.
Abstract
The counting function for the numbers satisfying the Collatz conjecture is studied. A related exponential congruence equation is investigated, yielding a method to construct its solutions from free variables, and enabling us to find at least Collatz numbers in the interval . The historical record is 0.84.
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Taxonomy
TopicsBenford’s Law and Fraud Detection · Probability and Statistical Research · Complex Systems and Dynamics