My friend and I were talking earlier today and he posed the following problem, that he does not know the answer to: Take a real number, $a$, and look at consecutive powers of $a$: $a,a^{2},a^{3},...$ and look at the fractional part of the powers, i.e. $a^k - \lfloor a^k \rfloor$. What values can the fractional part of the powers converge to, if any? Obviously $0$ is one fractional part, but he believes that all rationals can be. He then extended his answers to all numbers that are not transcendental. Just thought it was a nice question and I'm curious as to what the answer is myself.
Fractional Powers
9
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number-theory
sequences-and-series
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0I guess you are assuming that $a$ is not in $(-1,1)$, as that's the boring case... – 2011-11-18
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0We don't have to assume that, it just doesn't help answer the question; if $a$ is in (-1,1), the fractional part will be 0, which we already know is possible. I'm curious as to what other possible values we can get. – 2011-11-18
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0Some years ago I considered a similar question in the context of the collatz-problem. For some rational *a=b/c* I expressed the powers as irregular fractions and plotted the fractional part against the denominator and got some nice pictures. For the collatz-problem it was relevant, whether the dots for *1.5^N* are all below the main diagonal. But I found some nice structures for some common irrational numbers like golden ratio and pi so perhaps you'll find it interesting, too. Here it is: http://go.helms-net.de/math/collatz/aboutloop/GraphsOn3N_2NApproximations.htm – 2011-11-18