The unsolved number theory problem known as the 3x + 1 problem involves
sequences of positive integers generated more or less at random that seem to
always converge to 1. Here the connection between the first integer (n) and the last (m) of a 3x + 1 sequence is analyzed by
means of characteristic zero-one strings. This method is used to achieve some
progress on the 3x + 1 problem. In particular, the long-standing conjecture that nontrivial cycles do not exist is virtually
proved using probability theory.
References
[1]
Everett, C.J. (1977) Iteration of the Number Theoretic Function f(2n) = n, f(2n + 1) = 3n + 2. Advances in Mathematics, 25, 42-45. https://doi.org/10.1016/0001-8708(77)90087-1
[2]
Lagarias, J.C. (2010) The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society, Providence. https://doi.org/10.1090/mbk/078
[3]
Terras, R. (1976) A Stopping Time Problem on the Positive Integers. Acta. Arithmetica, 30, 241-252. https://doi.org/10.4064/aa-30-3-241-252
[4]
Crandall, R.E. (1978) On the 3x + 1 Problem. Mathematics of Computation, 32, 1281-1292. https://doi.org/10.2307/2006353
[5]
Halbeisen, L. and Hungerbühler, N. (1997) Optimal Bounds for the Length of Rational Collatz Cycles. Acta Arithmetica, 78, 227-239. https://doi.org/10.4064/aa-78-3-227-239
[6]
Eliahon, S. (1993) The 3x + 1 Problem: New Lower Bounds on Nontrivial Cycle Lengths. Discrete Mathematics, 118, 45-56. https://doi.org/10.1016/0012-365X(93)90052-U
[7]
Brox, T. (2000) Collatz Cycles with Few Descents. Acta Arithmetica, 92, 181-188. https://doi.org/10.4064/aa-92-2-181-188
