On the Verification and the Estimation for Two Expansion Problems of L. Collatz Conjecture

Abstract

[9] presented two expansions 5n+1 problem and 7n+1 problem for L. Collatz Conjecture which is an unsolved problem, and confirmed two expansions are correct for all n<100000 (=10^5). And [10] proposed two approximate formulas of the average number of steps for 5n+1 problem and 7n+1 problem respectively. In this paper, we confirm that two expansions are correct for all n<1000000000000 (=10^12), and propose two high precision formulas of the average number of steps for 5n+1 problem and 7n+1 problem respectively. Keywords: Collatz Conjecture, 5n+1 problem, 7n+1 problem, Verification, average number of steps.