An unsolved problem on the powers of \(3/2\). (English) Zbl 0155.09501
Let \(Z\) be the set of all positive numbers \(\alpha\) with the following property: “For every non-negative integer \(n\),
\[
0\le (3/2)^n\alpha -[(3/2)^n\alpha] < 1/2."
\]
This note makes an attempt to decide whether \(Z\) has, or has not, any elements, but arrives only at partial results. It is, in particular, proved that \(Z\) is at most enumerable; that every interval between consecutive integers contains at most one element of \(Z\); and that for sufficiently large \(x\) there are fewer than \(x^{3/4}\) elements of \(Z\) between \(0\) and \(x\).
Reviewer: Kurt Mahler (Canberra)
MSC:
11J54 | Small fractional parts of polynomials and generalizations |
Online Encyclopedia of Integer Sequences:
Numbers that are congruent to 0 or 2 mod 3.a(n) = floor( sqrt(2) * (3/2)^n ).
Floor( Pi * (3/2)^n ).
Number of steps to reach an integer == 1 (mod 4) when iterating the map n -> 3n/2 if n even or (3n+1)/2 if n odd.
Floor( e * (3/2)^n ).
Floor( phi * (3/2)^n ) where phi = (1+sqrt(5))/2.