How many numbers satisfy the \(3x+1\) conjecture?
Summary: Let \(\theta(x)\) be the number of numbers not exceeding \(x\) satisfy the \(3x+1\)-conjecture. We obtain a system of difference inequalities on functions closely related to \(\theta\). Solving this system in the simplest case, we establish \(\theta(x)>cx^{3/7}\). This improves a result of \textit{R. E. Crandall} [Math. Comput. 32, 1281--1292 (1978; Zbl 0395.10013)].
scientific article; zbMATH DE number 4122053