Lower bounds for Z-numbers
نویسندگان
چکیده
Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ > 0 is a Zp/q-number if {λ(p/q)n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Zp/q-numbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z3/2-number, then it is greater than 2 57. We also explore some connections between these problems and some questions regarding iterated maps on integers. 1. An approximate multiplication problem Let us begin with the following problem. Starting with a positive integer x, we consider the map x → ⎧⎪⎨ ⎪⎩ 4x/3, if x ≡ 0 (mod 3), (4x+ 1)/3, if x ≡ 2 (mod 3), STOP, if x ≡ 1 (mod 3). Consider the iterates of this map, starting, for instance, with x = 6. We have 6 → 8 → 11 → 15 → 20 → 27 → 36 → 48 → 64 → STOP. The reader might easily guess that the problem is the following: • Prove that, starting with any positive integer x, the sequence of iterates of this map terminates. Some simple computational experiments show that usually nothing exceptional happens with this sequence of iterates and it “behaves” properly. A simple heuristic argument suggests that a positive integer x “survives” at each step with probability 2/3, and so one would expect approximately (2/3)x of the possible starting values ≤ x to survive at least k iterations. It follows that the expected stopping time for an initial value x is likely to be bounded above by c log x, where c is some absolute constant. For example, the sequence starting with x = 3 terminates after k iterations. Although the first author has already stated this 4/3 problem at several conferences, it appears that there is currently no promising approach to its solution. This problem may very well remind the reader of the so-called 3x+ 1 problem, where the map is x → x/2 if x is even and x → (3x + 1)/2 if x is odd, and the Received by the editor January 22, 2008 and, in revised form, August 7, 2008. 2000 Mathematics Subject Classification. Primary 11K31; Secondary 11J71, 11Y35.
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ورودعنوان ژورنال:
- Math. Comput.
دوره 78 شماره
صفحات -
تاریخ انتشار 2009