The 3x+1 problem: new lower bounds on nontrivial cycle lengths
نویسنده
چکیده
Eliahou, S., The 3x+ 1 problem: new lower bounds on nontrivial cycle lengths, Discrete Mathematics 118 (1993) 45556. Let 7’: N -+ N be the function defined by T(n) = n/2 if n is even, T(n) = (3n + 1)/2 if n is odd. We show, among other things, that any nontrivial cyclic orbit under iteration of T must contain at least 17 087 915 elements.
منابع مشابه
A simple (inductive) proof for the non-existence of 2-cycles of the 3x + 1 problem
A 2-cycle of the 3x + 1 problem has two local odd minima x0 and x1 with xi = ai2 ki − 1. Such a cycle exists if and only if an integer solution exists of a diophantine system of equations in the coefficients ai. We derive a numerical lower bound for a0 · a1, based on Steiner’s proof for the non-existence of 1-cycles. We derive an analytical expression for an upper bound for a0 · a1 as a functio...
متن کاملA pr 2 00 2 Bounds for the 3 x + 1 Problem using Difference Inequalities
We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systems can be used to derive lower bounds for the number of integers below x for which the 3x+1 conjecture is true. Previous methods obtaining such lower bounds gave away some information in these inequalities; we give an improvement which (apparently) extracts full information from the in...
متن کاملTheoretical and computational bounds for m-cycles of the 3n + 1 problem
An m-cycle of the 3n+1-problem is defined as a periodic orbit with m local minima. In this article we derive lower and upper bounds for the cycle length and the elements of (hypothetical)m-cycles. In particular, we prove that there do not exist nontrivial m-cycles for 1 ≤ m ≤ 68. Our proofs are based on transcendental number theory, computational diophantine approximation techniques, and a not ...
متن کاملTwo Nonlinear Lower Bounds for On-Line Computations
One of the major goals of theoret ical computer science is proving nontrivial lower bounds on the (time or space) complexi ty of specific problems. Unfortunately, despite continued research effort for the last ten years, the success in proving lower bounds has been minimal. The only known general lower bounds are at least exponential [2, Chap. 1 1]. For no specific problem in NP can we prove a ...
متن کاملBenchmarks for Strictly Fundamental Cycle Bases
In the MINIMUM STRICTLY FUNDAMENTAL CYCLE BASIS (MSFCB) problem one is looking for a spanning tree such that the sum of the lengths of its induced fundamental circuits is minimum. We identify square planar grid graphs as being very challenging testbeds for the MSFCB. The best lower and upper bounds for this problem are due to Alon, Karp, Peleg, and West (1995) and to Amaldi et al. (2004). We im...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete Mathematics
دوره 118 شماره
صفحات -
تاریخ انتشار 1993