https://arxiv.org/pdf/math/0309224

164. Riho Terras (1976), A stopping time problem on the positive integers, Acta Arithmetica 30 (1976), 241–252. (MR 58 #27879). This is the first significant research paper to appear that deals directly with the 3x + 1 function. The 3x + 1 function was however the motivation for the paper Conway (1972). The main result of this paper was obtained independently and contemporaneously by Everett (1977). A positive integer n is said to have stopping time k if the k-th iterate T (k)(n) < n, and T (j)(n) ≥ n for 1 ≤ j < k. The author shows that the set of integers having stopping time k forms a set of congruence classes (mod 2k), minus a finite number of elements. He shows that the set of integers having a finite stopping time has natural density one. Some further details of this proof were supplied later in Terras (1979). This paper introduces the notion of the coefficient stopping time κ(n) of an integer n > 1. Write T (k)(n) = α(n)n + β(n) with α(n) = 3a(n) 2n , where a(n) is the number of iterates T (j)(n) ≡ 1(mod 2) with 0 ≤ j < k. Then κ(n) is defined to be the least k ≥ 1 such that T (k)(n) = α(n)n + β(n) has α(n) < 1, and κ(n) = ∞ if no such value exists. It is clear that κ(n) ≤ σ(n), where σ(n) is the stopping time of n. Terras formulates the Coefficient Stopping Time Conjecture, which asserts that κ(n) = σ(n) for all n ≥ 2. He proves this conjecture for all values κ(n) ≤ 2593. This can be done for κ(n) below a fixed bound by upper bounding β(n) and showing bounding κ(n) < 1 − δ for suitable δ and determining the maximal value 3l 2j < 1 possible with j below the given bound. The convergents of the continued fraction expansion of log2 3 play a role in determining the values of j that must be checked. 165. Riho Terras (1979), On the existence of a density, Acta Arithmetica 35 (1979), 101–102. (MR 80h:10066). This paper supplies additional details concerning the proof in Terras (1976) that the set of integers having an infinite stopping time has asymptotic density zero. The proof in Terras (1976) had been criticized by M¨oller (1978). 166. Bryan Thwaites (1985), My conjecture, Bull. Inst. Math. Appl. 21 (1985), 35–41. (MR86j:11022). The author states that he invented the 3x + 1 problem in 1952. He derives basic results about iterates ,and makes conjectures on the “average” behavior of trajectories. Note. The mathematical community generally credits L. Collatz as being the first to propose the (3x + 1)-problem, see Collatz (1986), and the comment of Shanks (1965). This would be an independent discovery of the problem. 167. Bryan Thwaites (1996), Two Conjectures, or how to win £ 1000, Math. Gazette 80 (1996), 35–36. One of the two conjectures is the 3x + 1-problem, for which the author offers the stated reward. 168. Robert Tijdeman (1972), Note on Mahler’s 3/2-problem, Det Kongelige Norske Vidensk- abers Selskab Skrifter No. 16, 1972, 4 pages. (Zbl. 227: 10025.) 62